I know some folks really hate microfiber, but for those of us who don't, I think that would be perfect as knitted or crocheted soap saver bags. Microfiber yarn is available in yarn stores and if one can find a very fine yarn (such as sock yarn), it would be perfect and dry extremely fast.
Several months ago, I bought some small microfiber cloths in the baby supply area at a store (I don't recall which one, as I was traveling) and they have replaced my terry cloth washcloths for rinsing the Noxzema off my face when I wash my face each morning. The microfiber cloths dry super fast, much faster than the terry cloth, when hanging on a hook in my bathroom. I would not expect them to dry as fast as tulle soap saver bags, but absolutely much faster than crocheted or knitted cotton does. Cotton takes way to long to dry without some added air circulation &/or heat in my bathroom anyway.