Digging through the knife suppliers, all the pieces of handle material are 3/8 or less. I'm looking for something a bit thicker for a project.

If your making a thick handle that is bolted together , you could always use 2 layers

When I ran into the same problem, I made my own.

Just take a bunch of strips of burlap/canvas/paper/linen....whatever you like.  (use 3x as much as you think you'll need because there's a lot of air trapped in the material)  Mix up some fiberglass resin, saturate each strip and layer.  Use a roller to squeeze out excess resin and air bubbles every few layers.  Sandwich the stack between a couple of scrap boards (use wax paper to keep it from sticking) and clamp tightly.  If the stack is sliding around and you can't keep it fairly straight when clamping, you have too much resin--use the roller again.

Let it set overnight and viola!  Homemade micarta.  It looks like shit at first, but it machines about the same as the commercial stuff and when you're done, you'd never know it was homemade.  You can make it from any material or color combination you want.  I've used colored construction paper, several kinds of fabric and even thin vinyl floor tiles with excellent results.  

You have to read closely but Jantz and probably Texas Knifemaker's Supply sell Dymondwood blocks that are 1 1/8" by 1 1/2" by 5".
This is stated at the very end of the description.  Blocks are listed for all the Dymondwood colors.

Just cut a block in half and you have two thicker scales:

http://www.knifemaking.com/product-p/pk201.htm