I've found a new series of books that are geared for those taking the JLPT.
http://www.whiterabbitpress.com/catalog/Japanese-Graded-Readers-Level-1-Volume-1-w-CD-p-16457.html
This link is a first in the series, and is for JLPT #4. Each has about 5 short stories that have to do with Japanese culture. They are very entertaining and also are easy to read. They have both kanji and kana, with furigana to the side of the kanji.
Volume 4 is actually for the people who are up to the JLPT level 2. I have bought all four volumes with the intent of being able to read everything. A word of warning though, these volumes read in the tradition Japanese direction of right to left, and up/down.
They also come with CDs so you can hear native speakers "read" to you.