[Archer82]

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May you'll like

Fiddler's Green

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AFAIR - The Real McKenzies and Pubcrawlers were pretty good.

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The Pogues
The Young Dubliners
McDermott's 2 Hours

Everyone of these is a great suggestion.  There are many more that I can't even think of off the top of my head.  Floggin Molly is my favorite band.  The Pogues are phenominal as well, make sure to listen to "Fairytales of New York" by The Pogues. 
 
EDIT: If you like a bluegrass feel check out William Elliott Whitmore.  He has opened for bands like Flogging Molly and went on tour with The Pogues.  Really popular with the Irish Punk scene for one reason or another, I happen to be a huge fan of him as well.  He sounds like a old man and is only in his early 30's he is beyond brilliant, just great soulful music.  His is not a punk singer or Irish yet has gathered a rather large following in those circles.

 

[Punnisher]

Another vote for the Frames. They are different from the Pogues or Flogging Molly though.

 

[Stoin]

My two favourite Irish bands:
 
Therapy?
 
Horslips
 
Slightly different to each other. Therapy? are a great mix of hardcore punk, metal, rock, industrial and something else. Horslips are just a great Celtic rock band - Trouble with a Capital T!

 

[duotone]

The Tossers
 
The Saw Doctors

 

[krmathis]

Yeah, how about some U2?

 

[R_burke]

Enter the haggis - especially their early stuff
 
Not quite so punk, but good is Oysterband
 
and of course the already mentioned
Pogues
Reel McKenzies
Young Dubliners