It really depends on what aspects of The Lord of the Rings you want to emulate or capture. The One Ring, for example, is very good at emulating travel, exploration, and the effects of evil influence -- key features of the original books -- but its abstract combat rules are perhaps not a good fit for the acrobatic fights of the films.
It would also be useful to know how familiar you are with the setting, or whether that's even a factor. The One Ring is focussed on the Mirkwood area and so if you're intending to adventure further afield, you'll need to come up with a lot of setting material yourself. Middle-Earth Role Playing has heaps of detailed setting material so you can let them do that work for you, but it's all out of print.
If you're very familiar with the setting -- I know people who have read and seen everything Tolkien related and are like encyclopaedias -- then I'd say find a game with a combat system you like and just adapt it to Tolkien's world. Anything could work; you could use D&D4 but limit players to the martial classes for example.
Otherwise, The One Ring is quite simple to learn and play and does a good job of emulating the feel of Tolkien's work, if you are happy with either focussing on Mirkwood or making up your own stuff, and you don't mind abstract combat.