You know, I'd never actually thought about that? Hmm....
I think the Nine Rings self-destructed when the One was destroyed. After all, those Rings were so totally in the thrall of the One (and probably drew so much of their power from it) that the end of the One Ring meant the end of the Nine. I could be completely wrong, of course, but it wouldn't be the first time...
EDIT: post number 42!! This is the Ultimate Post!