I
do use accents, sometimes, but only where [I think] they belong, and I'll tell you why ('cause I know you were just dying to ask

):
One thing that always struck me about Tolkien, even before I read an interview where he explains it, is that the further you get from "home" - i.e., the more foreign the speaker is to your "point of reference", the hobbits - the more archaic/dialect-y the speech gets. His intention was that you identify with the hobbits, as the most "pastoral modern English" culture, and that you got the same sense of awe and mystery about the strange foreign men and (especially) elves as the hobbits did. The idea was that
none of them were speaking English, but the accents and dialects used in place of their actual languages were selected to be familar or foreign, where appropriate.
In my games, depending on the setting, there's usually a "home" culture - the one the adventurers are from, or at least the most familiar with. Most folks from that culture, I portray as "without accent" (or, as accent-less as someone from New England can - luckily, there's no "cahs", they go to taverns instead of "bahs", and "wicked" has an entirely different connotation). I'll use a faux-cockney accent for people who are a bit "rough around the edges", bouncers, dock workers, etc. (no offense to anyone who has a cockney accent in real life, but that's the point of reference to a Yank), and a faux-"upper-class" English accent for people who are either of the nobility or university-educated. Dwarves (in a Tolkien-y stock fantasy setting) get thick Scottish burrs, elves have a hint of a British accent, and humans from foreign cultures get assigned a real-world accent, chosen based on a) real-world analogues of their homeland (those from a northern frontier will either get Scandinavian- or Slavic-sounding accents, for example), and b) my abillity to do the accent without stumbling over it.
I'm shying away from accents more and more, though, other than for a few chosen NPCs, because it interferes with my ability to come up with a distinctive voice for each major character - it's hard enough to play a voice low, high, fast, slow, with a lisp, a tic, whatever, when I then also have to overlay a regional accent on top of that. Alec Guinness I'm not.