The first game had Hayley Westenra singing mostly folky/classical songs of English or Maori origin.
http://en.wikipedia.org/wiki/Endless_Ocean#Music
I think the only one of Irish origin was The Last Rose of Summer

The second game featured Celtic Woman (inc Hayley Westenra) and there were a few Irish songs.
http://en.wikipedia.org/wiki/Endless_Ocean_2:_Adventures_of_the_Deep#Music
Besides Dulaman there was Carrickfergus:

And Isle of Innisfree:

Though my favourite song of EO2 is Nella Fantasia (which isn't Irish, nor folk):

I share your view on WIndwaker by the way, I loved the opening sequence... and each time I finish the game and it goes through the credits I end up being out of breath because I whistle it all the way through.

https://www.youtube.com/watch?v=9Vg-870E4ck

not sure how to embed but folklore has some of the best celtic music ever!

Creid, the Xenogears orchestal album, is said to be partly inspired by Celtic music.

http://www.youtube.com/watch?v=ycqNpRpB_tc&list=PL1688BEBE72E0E85E

There are at least three volumes of celtic arrangements of Suikoden music...