I’ve heard a lot of people talk about this, and I always wondered about it. Well, thanks to all of these people, I no longer have to wonder! Yay!
Here’s the collated story from their posts:
Why couldn't those giant eagles have just picked up Frodo with the ring and carried him to Mordor, you ask?
Simple, really. Remember the Nazgul? They could fly too. Sauron was scouring the skies as well as the seas and land. The only reason Frodo made it on foot was because of all the other distractions being laid by the wars and intentionally by his confederates.
Also, the eagles were servants of Manwe, of the Valar. Long before the War of the Ring, the Valar decided against getting involved with the wars of Middle Earth. However, sending an eagle after the ring was destroyed would not contradict that pact as it would be After-the-Fact, plus at that point Sauron and the Nazgul were powerless/destroyed, so the skies were safe.
Also, contrary to the way the movie tells the story, the eagles didn't carry Frodo and Sam all the way to Rivendell or Minis Tirith, they only took them as far as Ithilien and they went from there the rest of the way on horseback.
And there you have it.