By far the most fun type of mount to ride for most gamers is the dragon. Couldn't an inhabitant of Middle Earth find a jolly, friendly dragon to carry her wherever she wishes to go? No. It turns out that Tolkien's dragons are huge, evil fire-breathers who live in the northern areas of Middle Earth and are near extinction at the time the Lord of the Rings takes place. Even if our hero could find a dragon to ride, she would soon regret it -- the dragons aren't too keen on being controlled and will eat your elf for breakfast. No, the dragons of Middle Earth aren't friendly human-serving dragons. There is nothing they would like better than a fried hobbit on rye.
Aside from dragons, there are two other evil creatures in Middle Earth who have wings and are large enough to carry a person -- the Balrog, and the evil flying mounts of the Nazgul. Obviously it would be difficult to convincing a Balrog to carry you around. For one thing, there are only a few left in the world, and for another thing, it's not even clear whether they can fly, as there is no mention of it in the books. Of course, the flying steeds of the Nazgul can fly, but good luck prying one of them out from under their wraith-king masters. It's just not going to happen.
The mostly likely candidate for flying mount-hood in Middle Earth is the Eagle. These are not the small, lesser eagles -- these are gigantic, thinking, talking beings with names and customs. They have even been known to carry people from time to time. Gwaihir the Windlord rescued Gandalf from the Tower of Orthanc and flew him to Rohan, as well as rescuing Frodo and Sam from certain death in Mordor. So why can't they carry my chubby hobbit guardian over the Misty Mountains?
The answer is, well, why would they? Since Eagles are sentient beings with their own personalities, culture, and free-will, they wouldn't be keen on shipping our lazy behinds across the world at our beck and call. They were not made to serve humanoids like a horse or mule does. Even though the Eagles did occasionally carry people around, it was only in dire need, and usually at the request of Gandalf (who, being a sort of demigod in Middle Earth, had more influence than your simple man of Bree), and even then they constantly warned him that they were not beasts of burden -- they carried him because they wanted to, not because they had to.
Therefore, the idea of a tiny hobbit, brawny horseman from Rohan or smelly dwarf from Thorin's Hall controlling an Eagle and forcing them to fly at their bidding simply does not jibe with the wishes of Tolkien. So even if Turbine wished to implement flying mounts, it is almost certain that Tolkien Enterprises would disallow it for the reasons explained above.
Ignoring the lore, there is a much simpler reason why flying mounts are unlikely to be implemented : the terrain in the game is not designed to support flying mounts. Like many games, the terrain is built to look beautiful and finished only from where you can see it. If you look behind that stunning, tree-laden mountain, you'll find ... nothing. Turbine would have to redesign the majority of terrain in the game to implement flying mounts, and with so many people clamoring for more end-game content for the game, their time and money is best spent elsewhere.
The most simple reason, however, is that the developers have stated multiple times that they will not be implementing flying mounts in the future.
However, all is not lost. There is a possibility in the future that Eagles could be implemented as part of a quest-line, where you ride one on a set course as a one-time deal. This is purely speculation and is not supported by the developers at this time, but it is always a possibility for the future, so keep your fingers crossed!
In the meantime, do not fret about the lack of different mounts in LotRO! New mount styles are constantly being added, and in the book 13 update due at the end of the month there will be a new mount variant released. No further details are known, but in a recent chat it was announced that a new type of mount was planned, One that can "take you into places where horses would dare not go."