There aren't really any specific chord progressions that will say "Irish". The sound of the music is fundamentally based on melodic patterns; specifically turns and devices originally found in the technique of playing the uilleann pipes.
However, there is a pretty common progression in some of the jigs: Em and D. Try alternating these two while playing in a jig strum pattern in 8 bar phrases. For examples that use this progression, the Swallowtail Jig and the Road to Lisdoonvarna come to mind.
You might also try messing around with E-Dorian.

-Bryn

The tricky part about chording Irish music is that the genre was developed mainly on melody instruments that used a variety of natural scales and variants on the notes in them. This type of music doesn't lend itself well to structured chording.

Until the 1920 recording sessions Irish music was almost never played with chording instruments (the Uilleann Pipe regulators being one possible exception.) That leaves us modern chord instrument players with just a few choices: 20's piano style, which is mostly I IV V with a heavy bass note rhythm (contact Grey Larsen for more information about this), or a more modern style (a la John Doyle).

To get that more modern sound (in Major tunes, which usually follow a I IV V progression) you can try substituting minor cords. So a typical Irish progression that goes I IV V I I IV V I would work like I ii iii IV7 vi IV V I. or I ii iii IV7 I6 iii I. Any progression that creates a linear movement usually sounds quite good over Irish tunes. Just don't do it every time through.

In the Dorian or minor tunes, substituting in the III for the One sometimes works really well.

Here's an example of Crowley's Reel (from John Doyle).
Basic:
part A: D'''D'''D'''G'A'D'''D'''D'''G'A'
part B: D'G'A'''D'''G'A'D'G'A'''D'''G'A'
With Subs and passing chords"
part A: D'''D'''Bm'''Bm'''G(maj7)'G6'G6'''Em'''A'''
part B: D'Em'A'''Bfm'Bm'''G'A'D'Em'A'Bfm'Bm'''G'A'