Mulberry Street, by Dr. Suess. This is one of the good "Doctor's" earlier works. It is packed with fun and imagination that will also hold an adult's interest, which can be an important factor in implementation of the Rayment Learn to Read Method.
Harold and the Purple Crayon, by Crockett Johnson has some interesting art and concepts. All of the Crocket Johnson books are well executed.
It's Not Easy Being a Bunny, by Marilyn Sadler and illustrated by Roger Bollen, is a great book for this method because the animal names repeat periodically in the story.
Go, Dog. Go!, by P. D. Eastman is a good first or second book to try out the Rayment Method. Plenty of word repetition and lots of fun for the child.
Another favorite is "The Little Red Caboose" by Marian Potter. Unfortunately, this book currently seems to be out of print. However, you can try at Amazon.