As an update, your suggestions led me in the right direction. I went with a product called "wacky wood or wiggle wood." It's 3/16" thick and crazy flexible. Looks like it might be two layers of a very light (balsa?) wood on either side of a fabric. It makes a very fine dust when sawn but is really a nice product to work with. I got mine through a local store: Intermountain wood products
Next, I'll glue a veneer onto it to match my cabinets. Thanks again for the help.
More detail on my project blog: http://67streamline.blogspot.com