If you want to be precise, Peggy Gordon is from Canadian tradition, and seems to have been unknown in Ireland until commercially recorded in the late 1950s or early '60s; its roots are diffuse, but seem to owe most to Scotland and England (the tune is the well-known Banks of the Sweet Primroses), though doubtless there are Irish roots somewhere in there as well. Laddie (Lassie) Lie Near Me is Scottish; Maggie has been examined here in great detail in the past and is a product of the Canadian/American songwriting industry; both Song for Ireland and The Rose of Allandale are English. The last is particularly so, as the form in which it is usually sung derives directly from the Copper Family of Sussex, who made some characteristic and easily-recognisable changes to Jefferys and Nelson's original melody.
There is a strong tendency for the same songs to be mentioned every time anybody asks a very vague question like this. There are a great many very fine and interesting songs which are really Irish, and it baffles me why the same old recently-imported chestnuts constantly crop up. I suppose it's largely because of the omnivorous nature of popular performers like the Clancy Brothers, Dubliners and so on: they recorded anything that took their fancy (the Clancys are reputed to have paid someone to pick up likely material around the then-flourishing English folk clubs) and these songs were then learned from their records by a great many people who simply assumed, without thinking about it, that it was all Irish material.
The Cow that ate the Piper is a good thought. Beside the DT entry (ignore the query in line 2; the suggestion in brackets is wrong), see: