When I first saw How the Lord of the Rings Should Have Ended [1], I though, “Yeah, why didn't they use the giant eagles to fly the Ring to Mount Doom?”
One of the arguments I've heard is that the Nazgûls' fell beasts [2] would have taken out the giant Eagles once they entered into Mordor. But Sean Chist felt otherwise [3]—he thinks it's a plot hole left by Tolkien (link via Hacker News [4]).
But personally, I find this theory [5] much more plausible and a much better explanation for why “Operation: AirDrop One” wasn't done.
[1] http://www.youtube.com/watch?v=1yqVD0swvWU
[2] http://lotr.wikia.com/wiki/Fell_beast
[3] http://www.sean-crist.com/personal/pages/eagles/