There are actually many Rings of Power in the Lord of the Rings series. Though much of the story is focused on the quest to destroy the One Ring, there are other Rings that other beings possess. Nine were given to the men who became Sauron's Nazgül. Seven were given to the Dwarves. And the Elves possess three. Sauron played a role in the creation of the nine Rings given to men and the seven Rings given to the Dwarves, but not the three possessed by the Elves. So why did their power fade after the destruction of the One Ring if Sauron played no role in their creation?
To understand the answer to this question, it is crucial to know how the Rings of Power were created as well as their purposes. Sauron, of course, intended to use the One Ring to control the bearers of all the other Rings of Power. The elves, however, were too intelligent and once they sensed that Sauron had created the One Ring, took their Rings of Power off.
Sauron –who at that time was pretending to be Annatar, Lord of Gifts– was angry with the Elves. He waged war against them to get the Rings of Power back. He was able to find the nine that he gave to Men and the seven that he gave to the Dwarves, but was unable to find the three Elven rings. The Elven rings were made last, without Sauron's help, by Celebrimbor. The Rings were then given to Gil-galad, Cirdan, and Galadriel. At the time of the events of Lord of the Rings, Galadriel still possesses her Ring but the others are kept by Elrond and Gandalf. The three rings are Narya, the Ring of Fire, Nenya, the Ring of Water or Ring of Adamant, and Vilya, the Ring of Air. During the events of Lord of the Rings, Gandalf possesses the Ring of Fire, Galadriel possesses the Ring of Water, and Elrond possesses the Ring of Air. Each of the Elven Rings has its abilities, which faded with the destruction of the One Ring. Their bearers all went to the Undying Lands eventually.
Yet why is this, if Sauron never played a role in their creation? Well, there are two possible explanations. Firstly, while Sauron was still disguised as Annatar, he taught Celebrimbor and the Elven smiths how to make the Rings of Power. It is possible that therefore, Sauron, sneaky as he is, snuck in a condition that the Rings created would only be powerful as long as he was. He had been defeated before but since the One Ring still existed he was only temporarily defeated. This explains why the three Elven Rings would have been able to be used while Sauron was regaining his strength. Most of the story of Lord of the Rings, however, takes place in a world where Sauron is a very active threat. He is actively searching for his One Ring and is determined to control those who bear the other Rings of Power. Therefore, the three Elven Rings would retain their powers for a while, as long as Sauron was not fully defeated. There is also another possibility.
The word fade is an interesting one. It suggests that the three Elven Rings did not immediately lose their powers the second the One Ring fell into Mount Doom. Rather, the three Elven Rings gradually became less effective for their bearers, until they eventually would not work at all. This gradual loss explains why Gandalf, Elrond, and Galadriel did not immediately leave for the Undying Lands as soon as Sauron was defeated. The three of them had some time to spare and were able to finish whatever they wanted to finish on Middle Earth before leaving it. All of these characters are also powerful beings in their own right. All the Elven Rings did was amplify that power and possibly add some extra special powers. Therefore, the fading of the Elven Rings would not render any of these characters powerless. However, they all do eventually go to the Undying Lands, possibly due to the fading of the Rings, and possibly because they feel they've done all they wanted to do on Middle Earth. The Rings, in this case, would have lost their powers eventually due to the lack of the One Ring and possibly because they were designed to defeat evil, and evil, in the form of Sauron, had been defeated.
The Three Elven Rings served their purpose for a long time. Two out of three of them had several different bearers. It is also possible that the bearers of the Elven Rings at the end of Lord of the Rings know that it is no longer their world to live in. With Aragorn becoming king, a new age of Men has begun in Middle Earth. So perhaps it is no longer a place for Elves or Wizards. Or perhaps the Elven Rings would have faded anyway, with or without Sauron's defeat. However, it seems likely that Sauron might have made it so the Rings were somehow dependent on his power. In that case, the three Elven Rings would have faded because even though Sauron played no role in their creation, Celebrimbor did use what he learned from "Annatar", and perhaps Sauron, tricky as he was, snuck in a condition that kept the Rings dependent on him. Another possibility is that the three Elven Rings gradually lose their powers after Sauron's defeat due to the destruction of the One Ring and the fact that they are not truly needed anymore.
Of course, not every Lord of the Rings fan will agree with this assessment of the three Elven Rings. Some might think that if such a thing had happened, Celebrimbor would have noticed it. Or that the word fade does not necessarily mean a gradual loss of power. That's fine, everyone's different. Yet it seems like something that Sauron would do, and Celebrimbor, as good as he was, might not have noticed. He did make the decision to trust "Annatar" in the first place and to take his advice on how to make the Rings. So perhaps he wouldn't have noticed anything unusual at all. And if the Rings were rendered immediately powerless upon destruction of the One Ring, wouldn't Gandalf, Elrond, and Galadriel have left for the Undying Lands earlier?