The ring has been used as a symbol of unity long before the birth of christianity, so I certainly wouldn't discard your rings in some kind of anti-christian gesture.
Just because they are symbolic of a christian ceremony you took part in years ago I can't see any reason why you would want to discard them. I have worn a gold engagement ring for 4 years and have never seen it as being an issue as far as satanism is concerned.
It is just a physical representation of my commitment to my partner nothing more.
Man: An animal so lost in rapturous comtemplation of what he thinks he is as to overlook what he indubitably ought to be - Ambrose Bierce - The Devil's Dictionary.
Alice laughed. "There's no use trying," she said, "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." - Lewis Carroll, Through the looking glass.