LR is a system - a series of interconnected parts, which sum up to more. Nonetheless, here's some commentary on using various parts by themselves. It arose from the natural tension between individual exploration, personal taste, and trying to follow and understand a system.
LR revolves around the use of a parallel text, while listening to audio in the language you are trying to learn; the originator of the method, atamagaii, calls this 'stage 3'. Most of this document is about this stage, rather than the whole system. Generally, the parallel text is of a literary novel; it can originally have been in any language, including your native one or the one you're trying to learn (the original language makes some difference, but not much).
It's a gradual process. First, you start to get a sense of where you are in a paragraph/sentence; around the same time, you start to recognize cognates (including names) and get a feel for how the sounds shift between the languages. Then, other words start to become clear - first for a fraction of a second, with the parallel text, and gradually for longer, until you can passively recognize them in new contexts. Some grammatical constructs start to become obvious as well: structures for past vs present tense, the literary construction for '"Something", Ms. character said', etc. After a few dozen hours, going through a novel a couple times, almost every word will be at least transiently clear, in context and with the aid of the parallel text: the main exception are uncommon words, used in idioms that don't correspond closely to ones in your L1 text.
More complicated grammatical structures seem to take more time, especially ones that are particularly unfamiliar and notoriously hard to summarize or teach well: I've found aspect in Slavic languages quite tricky, for example. Exposure, careful observation, and accurate information from grammars go a long way; blindly believing incorrect or incomplete information from grammars can be quite frustrating - and counterproductive.
You can use a parallel text with audio in a language you already understand, to fill in gaps - you can learn unusual vocabulary this way, do intensive and extensive reading (see "The Art and Science of Learning Languages"), etc. This is what atamagaii calls "natural listening", and is essentially distinct from LR, which is a process used to arrive at natural listening. There is a bit of a continuum, though, as you can understand most things but still have gaps, which gradually get filled in as you learn and read and listen more. Using parallel texts and/or audiobooks at this stage is fairly uncontroversial; some polyglots highly recommend it.
You can use a parallel text with audio in a language where you're at an intermediate level. Perhaps you've studied it in a classroom for a year, or can read it but not understand it spoken, or speak a related language. The parallel text with audio is (part of) Listening Reading, and is a very quick and pleasant way to learn; your vocabulary will rapidly expand, as will your feel for the language.
You can use a parallel text with audio in a language where you have some basic knowledge: perhaps you know some of the most common words, and a little bit about the grammar, and it's in the same family as a language you already know - like German for an English speaker - or has a lot of shared vocabulary with a language you already know, like English and any of the Romance languages (French, Spanish, Italian, etc). This is harder; it can be frustrating at first, but becomes increasingly rewarding as you continue; your passive knowledge will quickly reach an intermediate level, at which point you can continue as if that was where you started.
And finally, you can use a parallel text with audio in a language that is utterly unfamiliar to you. This is a very controversial approach, but I like it.
First, a warning: jumping directly into stage 3 as an absolute or near-absolute beginner is not what atamagaii recommends. There are a lot of tools that can make your life much easier: doing stage 1 (reading the text in L1 beforehand; ideally, knowing it very well) and stage 2 (reading L2 while listening to L2, until you're sure you know where word boundaries are as you listen), using a text with a word-for-word translation into your L1, acquiring a knowledge of the phonetics, grammar, and script of your target language ahead of time, already speaking a closely related language, etc. Jumping directly to stage 3, unaided, is a very grasshoppery thing to do: you are, by definition, incredibly underprepared, not using LR as a complete system, and it is much harder than it would otherwise be. Using LR on a related language, or as an intermediate student is an entirely different experience. Nonetheless...
Jumping directly into stage 3 LR with literary novels works, even with unfamiliar scripts and distant languages. I've done it with Russian (an Indo-European language with a script vaguely similar to that of English); in about 40 hours, I got to the point where I could navigate websites and often get the gist of material on very familiar technical topics (cognates and context are very helpful); I mainly used "Crime and Punishment". In a handful of hours with Chinese, I got to the point where I could thoroughly analyze and understand a 7-character sentence as I heard it, with the help of a parallel text; I used "The Little Prince".
In both cases, I originally had to rely on cues, such as the length of audio files; when a file was nearly over, I knew I was on the last chapter of a paragraph, for instance. Having a narrator who speaks clearly is extremely helpful: ones who pause between sentences, and often pause a bit longer between paragraphs are invaluable if you're starting with an L1-L2 parallel novel, with no accommodations made for beginners. For Russian, names were very helpful; for the Chinese text I used, they were changed so much that they weren't at first - but they occurred often enough that they soon became a useful marker.
I recognized a few hundred Chinese characters from the start, due to previous study of Japanese (an unrelated language which borrowed part of its writing system from Chinese). For the first couple of hours, this was of no help whatsoever; I was too lost, and trying to process too much other information at once. After about 2-3 hours, it gradually became increasingly helpful, and I made some connections between sounds and characters as well, which in turn helped me ascribe meaning to what I was hearing from having an idea of the meaning of the characters.
I had very little time available when I experimented with grasshoppering into Chinese LR. Once I bootstrapped my comprehension from being utterly lost, to the familiar stage of being able to make out some common words, some grammatical structures, and even occasional short sentences, as well as being able to follow along with the Chinese text as I listened to the Chinese audio at the same time, I was satisfied; my goal was to bootstrap to a level I knew it would be easy to progress from, in the most unfamiliar language I had materials available for, not to get particularly far with Chinese. Despite the rougher beginning (due to a less familiar script, and a text and language with almost no cognates), I eventually found progress to be faster than with Russian; less inflected languages area easier to LR as a beginner, all else being anywhere near equal.
I would be very curious about the results someone would get by grasshoppering into a highly inflected agglutinative language, or a polysynthetic language, with no background in linguistic typology or previous knowledge of the grammar. I suspect it would be extremely difficult at first.
Regardless, this approach is jumping directly into the deep end. Any reasonable preparation work you do ahead of time will make it easier. My point is merely that it is an option which is open to you; you can choose to use LR as a system or grab parts as you wish, to do any or no preparation beforehand, etc. Effectiveness, difficulty, and pleasure will all vary; grasshoppering right to the center of the language may or may not be your cup of tea, and is certainly not systematic.
If you're already at natural listening to most of what you hear, pretty much any amount of filling in minor gaps with LR is helpful. If you've still got larger gaps, atamagaii has suggested a minimum of 2 hours a day, with easy texts and a language similar to one you already know; this sounds reasonable, but I haven't empirically verified it with a lot of people. If you're grasshoppering into the deep end, do it intensively; every extra hour in a day helps more than linearly, at least through 6, in my experience; atamagaii has written of doing 8-15 hours in a day, but I haven't tried that myself. At the earliest stages, a day off hurts a lot, and even a 20 minute break will leave you feeling like you're playing catch-up if you're trying something as far out as stage 3 LR of Chinese as an absolute beginner.
Experiment; I've found how much intensity is needed to be something which is pretty clear pretty quickly.
While natural listening with a parallel text, a tiny interruption (a friend says hi, your phone rings, you glance at a clock for 5 seconds...) is no big deal. With grasshoppering into the deep end, during the first few hours it can easily cost you 5-15 minutes, as you try to figure out where you were again, and perhaps have to restart an audio file and go back to the beginning of a chapter to reorient yourself at all. If you want to grasshopper into the deep end, you either need absolutely minimal interruptions, or even more patience and persistence than it would otherwise require.
Incredibly useful: knowing something about phonetics, grammar, and your native/strongest language. Knowing that phonetic contrasts that your L1 doesn't have are in your L2 is useful; being able to distinguish them is better yet. Some people seem to be able to do this instantly; for the rest of us, listening to minimal pairs and reading about how the sounds are produced and seeing lip and tongue images can help. Knowing that sounds are different in your L1 and L2 is essential; the boundaries of where an 's' ends and an 'sh' begins may be entirely different, and I think it helps you learn to hear correctly faster if you take this into account.
Having an outline of your L2 grammar can save you time, especially if it's markedly different from your L1. If you know it pervasively marks aspect, or uses particles to mark parts of speech, or folds pronouns for the direct object into the verb, you'll be less confused earlier.
Understanding your L1 is important; it's a useful tool for learning about phonetics and grammar. If you understand grammatical mood, have some idea of how your language forms relative clauses, etc, it makes learning these things about your L2 easier as well, and tools like reference grammars and phonetics charts more approachable.
I found reading a couple of books on linguistic typology, with morpheme-by-morpheme hyperliteral glosses of a wide variety of types of languages to be very useful; I did it before I heard of LR, and am glad I did. Similarly, a serious study of practical and L1 phonetics will benefit you every time you choose to learn a language.
Useful: cultural background about your target language, stage 1 and 2 of LR, knowing the script, gathering materials ahead of time. Cultural background helps you understand references and idioms - especially if your text was originally in your target language, or a culturally similar one. Knowing the name of the currency of the time the novel was written, or the name your L1 borrowed to describe the type of farming done (and which your L1 borrowed because it has no exact equivalent) can help more than you would expect: it's a word to recognize if you grasshopper in, deepens your understanding (and ideally your connection to the material) at any stage, etc.
Stage 1 and 2 of LR (mentioned briefly above) make the first few hours of stage 3 a lot easier; whether to do them is up to you, but I highly recommend it. Read the story in advance, and do stage 2 until you're comfortable with word boundaries and recognizing intonation cues - or don't, but expect stage 3 to take longer and be harder. You get stage 1 for free for your L3, L4, etc if you use the same texts, and for free in your L2 if you use a text you already know and hopefully love. If you skip stage 1, it can be harder to follow the story in L2 (especially in large, quickly-read, poorly aligned paragraphs) and easier to get caught up in reading L1 and ignoring the audio. If there's a golden rule of LR, it's "pay attention to your target language!" - especially the audio. Reading L1 while ignoring L2 is not L2 study.
Knowing the script: this will make your first few hours of study more comfortable. For a language with an alphabet or syllabary, a few hours to become familiar with the characters is reasonable; for Japanese, learn the kana (two syllabaries), for Chinese, consider Pinyin. Don't expect to be comfortable reading it immediately just because you sort of know the letters; that takes time. Personally, I've always skipped this step; it makes stage 2 or stage 3 LR harder when I start it, but I learn to read the script fairly quickly, and I dislike studying scripts in isolation, even with mnemonic tricks.
Gathering materials ahead of time: if you want to use LR systematically, to learn intensively and well, you don't want to have to scramble to gather materials midway through, or lack resources that can speed you up; get whatever set of parallel texts + audiobooks, grammar material, etc that you plan to use set up ahead of time. Atamagaii has a decent list of materials (insert link here).
If you already have some knowledge of the language, but poor pronunciation of it, initial vocabulary recognition and comprehension can be easier, but it'll slow you down overall: I find words I can't hear or pronounce correctly hard to remember, for instance. Unlearning bad pronunciation habits is also arduous.
I find it extremely difficult to remember words that I don't have a solid pronunciation for. There are two alternatives here: one is hearing the sound correctly, and the other is to invent a pronunciation for it. The latter is what you do by default if you read without listening, or learn from listening to people who share your L1 accent in the L2; both happen a lot in most classrooms and with most learning techniques.
The more effective way, in the long term, is to hear the sounds correctly; this makes listening to native speakers easier, distinguishing pairs of words in your target language easier, acquiring the intonation of your target language easier (which makes both speaking and listening easier, and even inflected grammar easier - a lot of details that are analyzed as irregularities are driven by phonetics/ease of pronunciation), etc. Listening is less tiring when you can hear the difference between words, rather than having to guess from context when you can't distinguish minimal pairs. As a bonus, it makes well-written literature even nicer, as you appreciate the euphony more; this is particularly apparent with poetry.
The quicker way in the short term is to use an invented pronunciation. This is hard to fix later; if you can reasonably avoid it, don't do it. If you only care about using a language for reading non-literary texts, or optimizing for the short term, it might be worthwhile - but that's a call for you to make, knowing the costs.
First, what are you trying to achieve? If you want reasonably good comprehension, I'm unaware of anything that rivals LR. If you want to visit a country where the L2 is spoken tomorrow, and you know nothing about the language, forget about it, learn a few phrases with bad-but-understandable pronunciation ('thank you', at a minimum), and have fun in the country as a tourist; LR will not serve you well. The best method is one that suits your goals, and that you can stick to (or effectively use as a supplement to other ones). If you hate novels, LR is probably a poor choice for you. If you hate ambiguity and not knowing what's going on and guesswork, LR might be handy for you to fill in gaps as an advanced student, but you'd be miserable grasshoppering into the deep end. Know yourself, and experiment. It's perfectly valid to choose not to use LR, or to use it as a supplement at some stage in your learning.
LR vs classroom study: with a good choice of novel, you'll only hear correctly-pronounced, grammatically correct input. You'll hear a lot of input: about a minute of native material per minute that you study, as opposed to perhaps a handful of minutes in an hour a few times a week. You won't be forced to produce grammatically and phonetically incorrect language and hear it for the sake of an exercise. You'll have a more eclectic selection of words and grammar at first: you'll quickly know some low-frequency cognate words and how quotations are structured, but may well not know how to say "my name is" or "thank you" the first day. Your study will be self-directed, at the intensity of your choosing, and without exams unless you go out of your way to sign up for them; whether this is helpful or a hindrance seems to be a matter of taste. In a bad classroom, without studying on your own, you may get nowhere; in a good classroom, you can learn quite a lot, but it will take you years, and you're likely to have to unlearn a lot too, especially pronunciation.
LR vs subtitled movies: no competition, LR is insanely more effective. LR features more words per minute, higher comprehensibility (a close literary translation, vs subtitles which are usually optimized to give the gist, compromised by factors like screen space and generally highly abridged), and less distractions (who's focusing on the details of the spoken words *and* subtitles during intensive action scenes, or when a lot is going on on-screen?). It's possible to learn through extensive watching; some people do, some don't (especially as beginners).
LR vs extensive L2-only reading: it's much harder to get started with L2-only reading unless the language is already close to one you know. A parallel text lets you have (slightly) comprehensible input from the first minute, and much more comprehensible input than you would otherwise have at the same stage as you progress. Many people advocate understanding at least 85% (or 90%, or 95%...) of the words on the page if you want to read in L2; it's not strictly necessary, but it makes it much easier. Another concern is if you're reading L2 without already being able to pronounce it well, or listening with an audiobook - you'll invent your own pronunciation of the language. It's not rare to come across Europeans who can read novels in a second language, but can't understand it spoken, who can only say a few words and phrases - and in a way that native speakers can't understand, or only with great difficulty. Fixing that is a lot of work. Extensive L2 reading is helpful - and when you're advanced in your L2 and want to improve further, vital - but it's inefficient and full of pitfalls for beginners.
LR vs one-on-one study with a tutor: the latter can give give you targeted production faster - in 10 hours with a tutor, you can learn a lot that's useful as a tourist (with a good tutor; with a poor one, you'll spend it drilling something useless, or hearing pointless rambling in your L1); in 10 hours of LR, you'll still be pretty lost unless your L2 is close to a language you already speak, or you already have a fair bit of knowledge of it. At 50 hours, you'll have likely learned far more words through LR. No one else can take responsibility for your learning; having a competent speaker you can ask questions can be helpful, though.
LR vs more conventional independent study methods: LR will bring you further, faster, if you can tolerate or enjoy the early stages, and if you could stick to either. If you can only stick to one or the other, pick that one. If you want to build from the ground up, rather than fill in from a high-level overview, you may prefer more conventional methods.
LR in conjunction with reference material: this is part of the system that atamagaii has written repeatedly about - a couple of pages of grammar cheat-sheets, good courses for the phonetics of your target language, pop-up dictionaries, etc are all part of the system. I personally like skimming reference grammars.
What do you want to do? How much time do you have available? How do you like to learn? How do you tolerate uncertainty? Do you prefer to drive your own study, or have it guided? The answers to these questions are quite individual, and have a direct impact on what methods make most sense for you at present. LR is not always the solution, or part of the solution; but for some people, it's a really handy tool.
Krashen is mainly known for advocating "i+1" comprehensible input. Parallel texts, plus previous familiarity with the text boost comprehensibility greatly. If you continue on with the audiobook, rather than stopping over details all the time, you get a lot of comprehensible input, at an amazingly high density per minute. I don't agree with everything Krashen has said, but I think LR is an amazing source of i+1 input; your brain picks it out as you go along. For intensive study, material at mixed levels (like in any novel I've ever seen) is a lot more digestible than graded input (like Assimil), in my experience. For non-intensive study (like the 20 or so minutes a day recommended for Assimil), it's the other way around - I'd recommend graded material.
One thing I've found useful is to skim reference grammars, textbooks, etc after a certain amount of LR. A textbook that could take months to work through can be mainly obvious after a few dozen hours of LR, and much of the rest is either artificial/wrong, or can be quickly absorbed by looking at a description, a few examples, and then doing more extensive reading. Exercises are an option too; I opt out. Regardless, I find material for beginners more pleasant to use a a quick pass to fill in any obvious gaps, after LR, rather than as preparation for LR; here's the idea in case it's helpful.