Hussar
Legend
Raven Crowking said:(1) Sauron would have noticed it, sent the Nazgul to intervene, and gotten the Ring.
(2) The person with the Ring (assuming that Frodo wasn't flying Air Gwahair) proclaims the Ring his before reaching Mordor. Sauron notices it, and sends the Nazgul.
(3) The person with the Ring proclaims the Ring his at the Fire, Gollum isn't there, Sauron notices it, and sends the Nazgul.
RC
Just to tweak RC's nose. Gandalf summons his pokemount Shadowfax, plops the ring in a saddlebag, sends it back to the celestial realms, turns to the left and incinerates Merry in a brilliant ball of fire, thus costing himself his summonable mount ability (replacement level from Dragon) and the ring resides forevermore out of the reach of Sauron. And the peasants rejoice.

On the idea of eagles. Well, it has been pointed out that we have this nice dark cloudcover, so, we load up Frodo on an eagle, send him out just below the cloudcover (not a chance he'll be spotted) and flies over Mt. Doom. Sauron doesn't notice it for the same reason he didn't notice it before - no one puts it on.
One of three things happen:
Frodo drops the ring, it falls into the lava, Sauron is defeated and the peasants rejoice.
Frodo proclaims the ring his, the giant eagle does a sharp bank, deposting both the halfling and the ring into Mt Doom, Sauron is defeated and the peasants rejoice.
Frodo proclaims the ring his, giant eagle folds his wings, dives at several hundred miles per hour, does an Ellen Ripley into the lava with Frodo and the Ring, Sauron is defeated and the peasants rejoice.
