What do you mean by trad? Some of those listed below are composed songs; ignore them if you want the "real" thing.

In the States, "The Bard of Armagh" (tune for "Streets of Laredo").
Black Velvet band (too well-known?)
Broad Black Brimmer
Come By The Hills
Dawning of the Day
Green Grow the Laurels
I Had a Hat When I Came In
Isn't It Grand, Boys (new, but a rouser of a chestnut)
Los San Patricios (Cris Pace)
Mingulay Boat Song
Mountains of Mourne
Sally Gardens
Shule Aroon
Spancil Hill