There are two books you can buy called "Read Real Japanese"- one is a collection of short stories, and the other is a collection of essays, each by contemporary writers. When I say "essays", I don't mean anything academic, one of them is about mens' attitudes to cooking, another is about telling little white lies- things like that.
I definitely recommend them for people who can read Kana and have a good grasp of basic grammar.
I think there are other books under the "Read Real Japanese" title but they're not laid out in the same way as these two are. I shall have to look into it.
Boths books have good introductions written by the translators, and there is also a picture and background information of each of the authors, also providing recommendations for other works of that author.
Basically, each book is in the Japanese format, where you read the book from the back cover to the front. On the right-hand page is the Japanese text, written in the traditional Japanese way- top to bottom, right to left. Every Kanji has furigana the first time you see it written.
On the left-hand page, is an English translation of more-or-less every sentence, but written piece-by-piece so that it's as close to the original Japanese as possible.
Now at the back of the book, written in the western style, first there is a dictionary with every word you find in the each of the stories/essays. After the dictionary, there is a grammar explanation of the main points of each sentence (every sentence except the very simple ones), and this section also provides a lot of information on Japanese writing styles- for example, why the author may have chosen to switch between the desu/masu style (which sounds subjective), to the plain da/dearu style (which sounds objective). Often there are explanations of why the "wa" particle is used instead of another, which is very useful.
Also, each book comes with a CD of each piece narrated by a Japanese actress. She does speak very fast at times, but in my opinion it's better to listen to Japanese spoken at a normal pace (i.e. very fast).
I find a great way to study is to sit at a computer, with the CD playing and the book (closed) with you. Try to write down each sentence, bit-by-bit, as you listen and skip-back the CD over and over. Then, once you've written out one sentence (you may recognise individual words or you may not), check in the book to make sure you heard each and every syllable correctly, and then, without using the book, try to work out what the sentence means, referring to the translation and grammar explanations when you've finished.
Another nice book is 'Reading Japanese With a Smile', which has nine articles from a wacky newspaper column. The article is in full kanji on the left page, with a translation on the right, and then a detailed section that breaks down each sentence with furigana, romaji, definitions, and reading and cultural notes. A couple of the titles are 'Son-In-Law of the Mob' and 'His First Call Girl Was the Girl Next Door'