I throw the wet mitt over the grab handle in the shower after I wring it out. It takes forever to dry. I do flip it over after a while. Dryer and hanging it outside are not options for me.
I strung up a short cloths line in the basement by the washer n dryer. I have a funky multi hanger thing I hang from it when drying many clothes or I hang my mitts from it too. Drys in a couple days down there...