There are almost no materials to read for absolute beginners. Even books for little children already expect understanding of fairly advanced grammar and quite rich vocabulary. While you might have studied for a few weeks or months now, those children have studied for years with native speakers.
Here’s a list of five websites that I’ve used in my beginnings. They contain JLPT N5, N4 and N3 reading materials.
UPDATE: Some of these websites aged away. For example the Chokochoko library is mostly gone. Check my other article about websites to read for more current list.
A collection of traditional children stories written mostly in hiragana with very basic kanji. They are also translated to English and accompanied by illustrations from books. Not only easy to read but these stories are an important part of Japanese culture. After reading these I understood plenty of references in Japanese pop culture. Even when I studied Japanese at school, the story called かさこじぞう was in my textbook.
UPDATE: Unfortunately the materials are no longer available, but backups exist!
Here you can find categorized articles and stories for all JLPT levels downloadable as PDFs. The articles don’t have furigana in the text, but difficult kanjis are printed again in the footnote with added furigana and explanations. Great way to learn to read and not rely on the furigana.
This page is for both English and Japanese learners. They have articles explaining English idioms in Japanese and some news articles that have added furigana (in brackets) worth reading. However they archive news articles only for premium members.
You can take a short version of the JLPT exam on the website itself and also download the official practice workbook from the year 2012 and sample questions from 2009 as PDFs. All of these contain the reading exercises as well.