I agree with MuseumGhost,
If the dishes are important to you have your renter sign an agreement and insist that he follow it or evict him.

I handled a similar situation poorly when I was younger (my bf and our male roommate refusing to wash dishes). After several what I'm sure were terribly reasonable requests on my part that they clean up after they cooked, the requests were ignored.

I'd come home from work totally steamed that the sink, which was empty and clean when I left, was piled with dirty dishes. So, I washed every dish in the house, packed them in boxes, and went to walmart and bought paper plates and plastic silverware. Problem solved. If they wanted to cook they could get a pan from the garage. If the pan was left dirty in the sink I threw it in the dumpster. Of course it helped considerably that all kitchen items belonged to said boyfriend. After the first couple of pans in the dumpster he magically learned how to do dishes.