I devised what I think is a good solution for the flat pieces, and one that makes them feel weighty, more substantial. I'm now mounting them on flat wood panels that I purchase from Blick and adding a 1/2" molding or frame of sorts cut from square dowels. My resident owner-of-a-miter-box showed me how to miter the corners. I sanded the edges, then fitted and glued the pieces to the panel. I mixed acrylic paint and painted the wooden edges with colors to match or enhance the felt colors. The panels are birch and pine, a good price, and come in a nice variety of sizes that happen to suit my pieces. This method satisfies me.
We learned years ago that you could hang textiles, like small rugs or woven wall hangings, from Velcro attached to a yardstick. The prickly hooks from the Velcro stick to the textile. I adapted that technique here. It turns out that you can buy a roll of just the hooks in either black or white at a craft supply store like JoAnn's. Because the felted work is not glued, but hangs from Velcro that is stuck to the panel, the felt can be gently taken out of its frame to vacuum or shake out, if necessary, or just handled.