Many wonder why the Great Eagles didn't just carry Gandalf and Frodo to Mt. Doom to destroy the ring, thus saving a lot of time and lives along the way.
In Middle-Earth, Tolkien describes the Eagles much like the origin of the Ents. They are beings without FÃ«a, or souls, if you will. Without FÃ«a, they are not constrained to do good and are not bound to the same fates as the other creations of Illuvatar (like Elves, Men, etc.). Thus they are self-serving, and like the Ents, could only have been constrained to join the battle out of self-preservation or anger (after destroying Isengard and imprisoning Saruman the Ents went back to their normal business, they wouldn't have marched on Mordor because they couldn't see the danger).
They were proud creatures, and do not enjoy carrying burdens, seeing themselves above the "pack animals" of the world, only occasionally helping Gandalf because Gwaihir, the Lord of the Eagles was once healed by Gandalf.
To help Gandalf by flying them across Middle-Earth would have likely been seen as unneeded, or below them. And they likely wouldn't have seen the urgency or necessity of the ring's destruction (few people outside of Gondor even acknowledged the danger of Sauron regaining power).
Tl;DR – The Eagles wouldn't have helped destroy the ring. So deal with it.
Stephen Colberts out there, feel free to correct any inaccuracies in this so I can fix them. I created this so that I could have a link to give people who try and point this out as a plot hole.
EDIT: In response to the questions regarding the theory presented in this article:
While a cool way of looking at it, you only need to reread the Council of Elrond chapter to learn that Gwaihir bears Gandalf to Rohan to get Shadowfax, not to the home of the Eagles to discuss a plan. The Eagles saw themselves as messengers, not as beasts of burden, so even then Gwaihir was like, Look Gandalf, I came as a messenger, not to carry burdens.
Also, Aragorn, Frodo, Gimli AND Legolas would have known who Gwaihir was. So all Gandalf would have had to say was "Oh guys, sorry I'm dying but Gwaihir is totally going to meet up with you, don't screw this uuuuuuuuuuuuup!" As he fell. Saying "Fly, you fools!" to explain all of that is stupid. Plus Gandalf tells them to "Fly" on multiple occasions before this point, but using the definition of "hurry your asses!" not the modifier "use the Eagles!" Sorry. Doesn't quite work.
Another reason why the Eagles wouldn't have worked (besides the nature of the Eagles), is that Sauron was watching the borders of Mordor quite well, and only didn't notice Frodo because they took the forgotten Morgul road, and he was distracted by the fact that he thought Aragorn had the ring. If they had flown Eagles into Mordor before the ring had been destroyed, they would have been swiftly killed by the foul beasts Sauron bred, or by the poisonous fumes in the air.
Read more: http://imgur.com/gallery/nMCHl