Wow, this was so easy and the results were so incredible, I thought I'd mention it here in case someone else hadn't heard about it.
I read about this method elsewhere, actually for cleaning oven racks without all the work. I thought I would try it for the stainless steel burner liners from my cooktop which I thought had been recently ruined, and I do mean ruined!
What I did, since ammonia fumes are so strong, is to place the burner liners in a glass mixing bowl, poured in enough ammonia to cover them, then covered the mixing bowl with a piece of plastic cling wrap. Left for about 24 hours, took them out, ran water over them and lightly swiped them off with a plastic-bristle brush and they look like brand new! :D I wish now that I had taken before and after pictures!