I dont, usually just fill up the sink with soapy water, rinse well and place them between paper towels to dry on the counter. Mine have become quite greasy over the years though, so I'm wondering if you can, but I dont think so.
Maybe their so greasy because of the food network chefs using them, my family (husband, brother, friends) come over and recreate iron chef in my kitchen and dont clean them as well...
I wouldn't put them in the dishwasher. Hand wash them and towel dry them and you should be fine. 
don't put in dishwasher...instructions say to just wash with warm soapy water.
I have put my silpat in the dishwasher and they come out just fine. I put them in the plate slot and round the edge into the plate slot next to it. Works just fine and the come out squeeky clean Just be careful not to crease the silpat or you will run the risk of cracking the fibers inside the silicone.
I don't think you are supposed to put it in the dishwasher. But if Ilovemyboys did and it came out ok then maybe.... I might put it in just once to get that greasy film off. I too can't seem to get it SHINEY and clean (yeah, i know that was a bad pun, shiney).
Because the weave inside the Silpat is glass, the instructions emphasize NOT washing in a dishwasher. As well, they are supposed to be stored flat and not rolled. Both reasons are so the glass weave does not become stressed and cause breakage within --rendering the performance of a Silpat moot.
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