I believe I remember hearing they got destroyed or taken by dragons

Yes, it says 4 of the 7 were consumed by dragon fire and that the the remaining 3 were captured by Sauron, but just wanted to know if it states anywhere what happened to them after Sauron is defeated.

I believe that if it was not taken by dragons, it would be somewhere up in the Withered Hearth, as when the dragons attacked, it most likely got lost in the chaos.

I don't recall it being stated anywhere what happened to the remaining three rings, but I think it's safe to assume that they were destroyed with the downfall of Sauron.

Four of the Seven were destroyed/eaten by dragons in the Dragon Wars against the Dwarves. The three remaining were gathered to Sauron in an unknown period, most likely between the Hobbit and Lord of the Rings.

IIUC, the rings can be divided into three categories.

• The One was made by Sauron without Elvish participation.

• The Nine and Seven were made by elves and Sauron working collaboratively. Some of Sauron's code was channelled through the One, so that stops working. It's enough to destroy the Nazgul. Does it completely destroy the Nine and Seven? Who can say. It's an interesting thought that these rings might still be floating about in the Fourth Age, no longer functioning the way Sauron intended but maybe not the way Feanor did either

• The Three were made by elves, with Sauron teaching and perhaps suggesting but not actually crafting. Destroying the One might have no effect at all.

^

Celebrimor, not Feanor