I bought 1/2-inch mesh from Home Depot (the kind used for fencing) and cut it down to the right size. I probably could have had it cut for me, but I wasn't sure of the size I wanted until I got it home.
I threaded ribbon through the outside border to finish it off a bit. Most pieces are hanging on S-hooks, though some earrings are just hooked through the mesh.
All in all, the total cost was less than $15 (mesh, ribbon, and S-hooks).