Well, a lot of "Irish" songs sung in the US, actually came from Tin Pan Alley. But most of those songs the Clancy Brothers and Dubliners and other Irish acts sang, actually came from Ireland - but much later than the Irish immigration.

Older songs? How about Kathleen Mavourneen, or the songs of Thomas Moore or Percy French, or Star of the County Down. Too many to count or list, I think.

-Joe-