By that standard any song not in Gaelic(Irish) is not really Irish. Give it up. The Irish musical tradition is very inclusive.

How about:

I'll tell my Ma Star of the County Down The Lakes of Ponchetrain (Ok so it's not irish- shoot me) The Water is Wide The Briar and The Rose (Tom Waits but sounds very Irish)

The book "Rise up Singing" Edited by Peter Blood and Annie Patterson has most of these songs. It's a great reference that includes many traditions.