Try Harry Potter. I've looked at it and would say it's lower intermediate. If you're around that level, try some novels aimed at young teens/high schoolers.
I know of two great books that can give a good jump into Japanese literature for anyone.
The first is called: Breaking Into Japanese Literature by Giles Murray. It takes stories by Natusme Soseki and Akutagawa Ryunosuke, presents the original Japanese with furigana the top half of one page, an English translation on the next page, and vocabulary on the bottom. I found this to be an invaluable book to begin reading. (You can also download a track of someone reading it online to practice listening). I would whole-heartedly recommend this book.
The next is called Read Real Japanese by Janet Ashby. It takes essays by some modern Japanese writers and presents notes on them. It doesn't give you a word-by-word translation, but does give you thorough notes. While I would say this one might be kind of hard, it's a good challenge.
If you think you don't need any more help and want to completely dive in, go for something on the best-seller's rack; odds are most of those things won't be too dense if the Japanese public is anything like the American public.
Right now, I'm reading a book called いま、会いにゆきます by 市川拓司 (Ichikawa Takuji). Don't tell any of my friends I'm reading it
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but it's a love story about a man who loses his wife and has to raise his young son on his own. However, before his wife died, she told him that when the rainy season came again next year, she would return to be with them. Now a year later, father and son go for a walk on the first day of the rainy season and find a woman in the forest who looks exactly like the wife/mother they lost. Naturally, she's lost her memory, so the man has to tell her the story of how they met and fell in love and they are all able to have a little more time together and find themselves.
I know. I know. It sounds like sappy love-crap. But my Japanese friend recommended the movie to me, which I, for some insane, non-masculine reason, absolutely adore, and then said I should read the book. The book is extremly easy to follow: I'm 150 pages in in about 2 months of casual reading while I've finished up school, and aside from needing to look up a ton of vocab, there've only been about five times I've really needed to stop and try to figure a sentence out. Plus, the book's different enough from the movie to keep things interesting.
Anyway, easy read, good movie, and the girls think you're a sensitive sweetheart if they catch you reading it (I'm planning to use it extensively when I get to Japan!)
Let us know how it goes.