How to take care of Silicone cookware
You're doing all you can. When I first started using Silpats (love them) I noticed the greasy feeling after washing. I googled and googled but all I ever found was that this just seemed to be a fact of life. I personally wash mine in the sink, and I think it turns out better, but I wouldn't be surprised if it's just my imagination.

From what I can tell it does not affect the quality of my food. I'm also pretty sure that it's not "grease" per se, but probably just some feature of washing and using silicone regularly.