A trick to prevent the loud thumpity thump of shoes in a clothes dryer is to hang the shoes inside the dryer door by hanging the shoe laces outside of the door. You tie the shoe laces into a knot to keep the shoe from dropping down. Noah shows how it’s done here.
What if your cycling shoes use straps instead of shoe laces? I used a small skirt hanger and clipped the straps to that, like so.
Close the dryer door onto the straps so that hanger hangs outside of the dryer, while the shoes sit inside.
This is an expensive way to dry shoes — I normally stuff paper into the shoes — but it gets the job done quickly.