The Flying Saucer
in Raleigh has "Trivia Bowl" aka "Think While You Drink" on Tuesdays. There are two rounds at 7 and 9:30pm where they have over $200 in prizes. I've been in a winning group before and got some free pitchers out of it.
Sammy's Tap & Grill
over by State in Raleigh has trivia on Tuesday nights as well at 8pm. You might want to avoid it however depending on how you feel about college students.
(http://nappertandysirishpub.com/raleigh/raleigh.html - broken link) in downtown Raleigh has trivia on Tuesday nights, but they call it "game show night" with the chance to win a "$1000 prize".