Well, you're in luck.
There's a bunch
The current, in print, RPG is called The One Ring, made by a British company called Cubicle 7 and written by the guy who wrote the Lord of the Rings boardgames. It's currently focussed on the region around Dale in the aftermath of the Hobbit.
The is a well regarded game, which draws heavily on the peculiarities of Middle-earth.
The next contender is the Lord of the Rings Roleplaying Game, by Decipher, which came out in 2002. Out of print, but can be found floating around on sites like scribd.com. It's not perfect, but its a beautiful book, which draws heavily on the flavour of the films. It's all about big rolling battles and so on.
The last contender is the oldest of the lot, Middle-earth Roleplaying, by Iron Crown Enterprises. MERP is the grand-daddy of them all, with hundreds of supplements. MERP is an outgrowth of a game called Rolemaster, which is itself a set of house rules for Advanced Dungeons and Dragons.
There's also a spin off of MERP called the Lord of the Ring Adventure game, which is probably the easiest of the three to play and the most fun, though rather simple and simplistic.
Hope this helps as a starter.