There was a recent post on this very subject. It was suggested that you place the rings in a plastic bag that has ammonia in it. Let them sit over night and the baked on grease and grime should come off easy.
Yes, the ammonia in the bag trick works. My wife found that on Pinterest and she used it on the burner grates. It worked great! I'm not sure how it works on the chrome drip pans.. if it damages the finish or not.