I would suggest "The Isle of Innisfree", some would say as popular as "Danny Boy". It is a truly beautiful song about Ireland and one which was written by an Irishman in 1949 so it would not be a traditional.
If you go to the thread "Isle of Innisfree" you will find a couple of great postings by Gerard Farrelly who is the son of the man who wrote "Innisfree".
PS There are two versions of "Galway Bay", one written by an Englishman which is the popular one as made famous by Bing Crosby and the more traditional "Galway Bay" which was written by a man from Kinvara in south galway, I cannot remember his name.
Love the site.