The American Dental Association recommends rinsing your toothbrush with tap water after you use it, then letting it air-dry. Keeping it in a closed container creates an ideal moist environment for the growth of bacteria.
According to the ADA, "there is no clinical evidence that soaking a toothbrush in an antibacterial mouth rinse or using a commercially available toothbrush sanitizer has any positive or negative effect on health." And using a dishwasher or a microwave could damage your toothbrush.
Experts recommend following similar advice for retainers or other dental devices that you put in your mouth, although additional disinfection with a denture cleanser may be recommended. Ask your dentist about the proper way to care for any particular dental devices that you may be using.