I do, both when I DM and when I play.
It's fun, it's amusing, and it differentiates what I'm saying from what my character is saying.
In the D&D game I play in, my character has some kind of Irish/Scottish hybrid accent. (It's slightly to the left of Ewan Macgregor in Trainspotting, and not really very accurate, but people laugh, so what are you going to do?)
In the M&M game I run, I try to give the villains notable accents to make them recognizable. I believe I actually got a blush out of one of my players when she tossed out an insult to my trickster demon, and I responded (in a decent Ricardo Montalbon impersonation), "Ahhh, (heroname)... your acid tongue is the perfect complement to your beauty, like the hint of dryness in a glass of fine wine."
(I should also note that when I say "I do accents", what I actually mean is "I do bad-to-decent impersonations of notable celebrities who HAVE accents". I can't do Castillian or Scottish or German. I can do Ricardo Montalbon or Sean Connery or 'Stereotypical Overblown Nazi in old WW2 movie', though. Not well, but recognizably.)
Also, the leader of the assassin's guild in my big D&D campaign talked like Jimmy James from Newsradio. That was kinda fun. I just had to start doing the accent, and people knew who they'd encountered.