[Zigûr]

Originally Posted by Faramir Jones

Ever since I first read LotR, I've thought that it would have been very easy for anyone, not just Sauron, to figure out who those bearers might be. They would have to be prominent Elves, with the ability to wield the rings properly; so a simple process of elimination would narrow down the group of possible candidates.

I think so as well. To quote myself from a while back (nearly 2 years ago in fact!),

Originally Posted by Zigûr

Logically speaking, there were only three places where the Three could be kept: Rivendell, Lórien and Mithlond - anywhere in which the Noldor still lingered. Sauron didn't know where the Shire was until Saruman told the Lord of the Nazgûl, which suggests to me that he had not necessarily spied out the West sufficiently to know, for instance, that Círdan no longer possessed one of the Three.
My point is that, according to Sauron's wisdom, to whom would the Three have logically been allocated? Lórien was the closest to both Moria and Dol Guldur, and resistant to assault, which implies a Ring being present at that location. If Sauron knew or at least suspected that Gil-Galad held some of the Rings prior to his death, I think it would make sense, by his logic, that he had passed the greatest of them, Vilya, to one of his subordinates - Elrond or Círdan. That leaves Narya and Nenya to be accounted for.
Despite the fact that he himself did not have a hand in creating them we could also imagine that Sauron was aware of the respective properties of the Three. Depending on the circumstances, this may have led him to at least be able to take an educated guess as to Nenya's location.

Originally Posted by jallanite

This implies, though does not prove, that Gandalf here uses “the Rings of Power” as a synonym for the Elven-rings. Gandalf could be conceived as jumping ahead to the Sauronic Rings in his explanation.

Yes I think in this case it could conceivably be argued that when Gandalf says "the Great Rings, the Rings of Power" he is associating the term "Rings of Power" with the term "Great Rings", the Seven and the Nine, specifically.

Originally Posted by jallanite

Thrór giving his Sauronic Ring freely to Thráin is a difficulty. Gandalf must be conceived of not to know that Thrór gave the Ring to Thráin freely. That is a possibility I admit, though it seems to me unlikely. But unlikely possibilities occur in real life. In “The Council of Elrond” Tolkien makes Gandalf say openly:

Thrór gave it [the Ring] to Thráin his son, but not Thráin to Thorin. It was taken with torment from Thráin in the dungeons of Dol Guldur.

In short, if Gandalf did not know that Thrór gave his Ring freely to his son Thráin, and if the term “Rings of Power” is used at times specifically for the Sauronic Rings as opposed to the Three Elven Rings, then Gandalf is telling the truth.

Perhaps Gandalf knew or assumed that, as a Dwarf, Thrór would be more resistant to some of the effects of the Ring, and thus more capable of passing it on to his son?
It could be argued that when Gandalf says to Frodo that no one ever gives up a Ring freely, he's specifically talking about Men (and, by extension, Hobbits).
I realise this is narrowing things down a lot but at the same time it does seem to suggest that he's giving Frodo the information which is most relevant to his particular situation.
It might be that Gandalf's situation is that of a teacher trying to introduce a complex point of lore to a student unfamiliar with the topic: not giving them extraneous, but more accurate, information, if he thinks it will confuse or distract the student.
Or, of course, it could just be Professor Tolkien making a mistake or not worrying about the readers themselves needing too much accuracy at that point.