You can’t put sushi mats into dishwasher, since they are made of bamboo. Covering the mats in plastic before rolling maki rolls is also a bad idea: plastic film is slippery to grasp and doesn’t drain excess moisture from the maki.
The best way to clean out the rice is to rinse the matt in hot water and leave to dry. Dried rice will shrink back in size, loosing most of its sticky wax. Now just wash the used matt in soapy water, and it should be clean as new.