I put my shoes in the sun all the time. Dries them out and kills the stink. If they're really muddy, I'll hose them down, but I've never bothered to wash a pair of running shoes--they'll just get all messed up again anyway.

There's a lot of conventional wisdom out there about the best way to care for running shoes. In my experience, none of it makes much difference. Pretty much no matter what I do--whether or not I rotate my shoes, or wash them, or leave them in the sun, or expose them to direct heat, run in them when they're wet--nothing affects how long my running shoes last except one thing; running in them. It turns out running in your running shoes is abut the worst thing you can do to them if you want them to last a long time. No matter how well or how badly I care for them, I pretty much get the same number of miles out of a pair of shoes (500-600 depending on model) and when they're shot, they're shot.