Westlife and Boyzone.
One of my best friends is from Galway and she goes on about how good the Irish are at music. Judging by most of the evidence, this isn't true. There are only a few good ones I could suggest from memory.
The Pogues and Damien Rice (not a band, but hey). Van Morrison (not a band, but good all the same) and ASH are also good if you're including the Northern Irish.
The Cranberries are ok too, although they're probably easier to remember here for sounding very Irish rather than being good.