If you are interested in irish traditional music, you should come to Piper Down on Monday nights. Every week starting at about 7 PM, musicians and punters converge for a night of tunes and "craic" [fun]. Come to play or just to listen, but don't miss it!
As I don't live here anymore and can't confirm that the sessions continue, it'd be a good idea to ring ahead and make sure the music is on before you show up.
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EDITED August 2006: I am informed by a friend who still lives in Salt Lake that the music has moved just up the street to a pub called The Republican. It's still every Monday night.