What happened to the power of the Rings once the One Ruling Ring was destroyed?

The most widely held position is that with the destruction of the One Ring all the remaining rings also lost their power (including the Three Elven Rings) and became ineffectual against the passage of time for which they were created.  The reasoning behind this is because the One Ring was embodied with the power necessary to bind and control all the Rings of Power when the One's power was destroyed so was the power of all the other Rings.  Those who held the Three; Gandalf, Elrond, and Galadriel departed over the sea into the West and took the Rings with them.

The evidence for this is found in several places:

"Yet many voices were heard among the Elves foreboding that, if Sauron should come again, then either he would find the Ruling Ring that was lost, or at the best his enemies would discover it and destroy it; but in either chance the powers of the Three must then fail and all things maintained by them must fade, and so the Elves should pass into the twilight and the Dominion of Men begin."

[The Silmarillion]

  

"But when all these things were done, and the Heir of Isildur had taken up the lordship of Men, and the dominion of the West had passed to him, then it was made plain that the power of the Three Rings also was ended, and to the Firstborn the world grew old and grey."

[The Silmarillion]

  

"But what would happen, if the Ruling Ring were destroyed, as you counsel? asked Gloin.  We know not for certain, answered Elrond sadly.  Some hope that the Three Rings, which Sauron has never touched, would then become free, and their rulers might heal the hurts of the world that he has wrought.  But maybe when the One has gone, the Three will fail, and many fair things will fade and be forgotten.  That is my belief."

[The Fellowship of the Ring]

The most melancholy reference is when the lady Galadriel explains to Frodo the fate of the Elves upon the outcome of the quest:

"Do you not see now wherefore your coming is to us as the footstep of Doom?  For if you fail, then we are laid bare to the Enemy.  Yet if you succeed, then our power is diminished, and Lothlórien will fade, and the tides of Time will sweep it away."

[The Fellowship of the Ring]

Because the primary power of the Three Rings is to slow or arrest the passage of time, Galadriel's words clearly mean that with the destruction of the One Ring, they too lose their power.