The Unicorn is as Irish as Shel O'Silverstein, but the Irish Rovers did it, so that makes it Irish. (No complaints from me, really; I love to do it, and there are great movements to go with it -- depending on your audience's age bracket!!)Don't forget Tommy Makem's "Johnny Is a Rovin' Blade".
Want a really pretty one? "Cliffs of Dooneen" is really picturesque.
Want to gross them out? "Follow Me Up to Carlow" is pretty gruesome.
Have fun, Alice!