If you really don’t want to put your kid’s shoes in the dryer at all, dry them overnight. Loosen the shoe strings really well, and pull the tongue all the way back. If you don’t have all night to dry the shoes, you can use a blow dryer on them, put them near your fireplace, or put them in the oven on a lower tempurature (that’s what we did 15-20 years ago when i was a kid). Also putting them outside in the sun is good if you are able to do that. It helps freshen your sneakers and saves energy.

Finally, in Woman’s Day magazine, there is a note about Sport Sox, 8-inch cedar-filled pouches that freshen shoes and can dry out rained-on shoes in two hours. They are 17.95 at cedaramerica.com.
References :
Womans Day Magazine about the Sport Sox