Use a mixer that buys your dirty coins and gives you clean coins in return, for a small commission.
It sounds counter-intuitive, but it works!
But the question is, how do you know the coins are "clean"? Who do you trust to tell you that the coins are "clean"?
You can insert the destination address into any AML checker, free or paid, for proof. Those services usually use the same blockchain analysis engines as commercial products. The disadvantage is that your address is stored in a private record forever, but that is probably irrelevant if it's only been used for receiving mixed coins.