Despite the enormous volume of Japanese text available on the internet, it can sometimes be a bit of a challenge to find something both interesting and level-appropriate. The following links should help you to find materials for your extensive reading.
Literature
The mothership. This is the Japanese answer to Project Gutenberg. Aozora Bunko can keep you occupied for a lifetime. I recommend starting with the works of 新美南吉 (Niimi Nankichi), a prolific children’s author.
A large collection of graded materials, nicely laid out with the definitions in a sidebar. Also has useful tools and links.
Reading Tutor’s Level Analyzer
One of the best things about Reading Tutor. Paste your text into the box and it will tell you approximately what level it is, in JLPT gradations.
A treasure trove of parallel texts. These are mostly English works that have been translated. Some non-fiction as well, but mostly classic works like Alice in Wonderland, Sherlock Holmes, and so on.
Legal free downloads of manga. Ad-supported, so click on some ads while you’re there and help make this project a success.
A large collection of classic stories, both Japanese and English originals. Has audio, transcripts, and parallel texts.
A delightful collection of Japanese folk tales, with very simple text, and fun illustrations.
Genpaku Project (list of works)
A large repository of translated works. Everything from Alice in Wonderland to Sun Tzu to the GNU license.
The good Squire Starsquid has collected quite a few articles from web sources, and arranged them by grade level.
Many easy stories, with audio as well. Thanks to ヤナ for the reminder!
A site for amateur authors to submit short stories of approximately a thousand characters.
Subtitles and Lyrics
Reading along with your favourite songs, drama, or anime can be very helpful to develop reading speed and listening comprehension. I don’t classify it quite as tadoku, since the fixed speed means you have no time to ponder over meanings of words or expressions, but it’s equally useful in its own way. Here’s some resources that should last you quite a while.
If Japanese subtitles for anime exist, this is where you’re likely to find them.
D-addicts Japanese subtitles index
As above, for dramas.
The best site for lyrics. There are others – Goo is probably more popular – but J-lyric is the only one I’ve found that lets you do anything with the text. Goo and others display the lyrics in a generated image file (and also disable right-clicking to discourage you from copying it), but J-lyric displays them in plaintext so you can put them in your SRS or Learning With Texts or whatever else you like.
If you know of other links along similar lines that should be added here, just leave them in the comments and I’ll add them. Thanks!
Lan'dorien's Mysterious Journey 
Awesome list. I didn’t know about a few of these. Thanks!
Very helpful list! I’m gonna have to read through some of those Japanese stories later when I get some free time (o´ω｀o)ぅふふ
yay resources! I actually am using Reading Tutor as regularly graded homework based on the questions and summaries in my Japanese class~ :p
Rikaichan Firefox addon is a popup dictionary. Shows reading and enlish meaning of single kanji as well as compounds and kana-only words. Very useful tool.
The port for Chromium (Google Chrome) is Rikaikun.
There is now a further development of Rikaichan called Rikaisama, which among other things has the very useful options of audio and Japanese definitions (from Sanseido).
http://rikaisama.sourceforge.net/
I agree, these extensions are extremely helpful for reading anything online.
http://hukumusume.com/douwa/pc/jap/index.html this has folklores with audio too. The readers are pretty good.