The obviously place is Lord of the Rings Online.
It's free to play, but much of the later game is behind a pay-wall. Still, you can have some good fun role-playing with people at the mid-levels. If you want to play, and want to RP, make sure you choose a server that ends in "-RE" (Roleplaying Emphasis) or "-RP" (Roleplaying), where most of the good RP happens. Some of the other servers don't RP at all. I've got a character on Landroval-RE and Laurelin-RP. The former is by far the more active, but the time of encounters I have with players on Laurelin are usually better.
Beware of snobs who've never read Tolkien, but will scour wikipedia to find some reason why there's a problem with your chara's back story.