I happened to see both "The Quiet Man" and "The Searchers" shown at the annual John Wayne Film Festival in Newport Beach three years ago on the 100th anniversary of his birth. The panel discussion included Patrick Wayne and John Ford's grandson. IMDb.com now has much better information on soundtracks often including where the music can be found in the film. For example, it states that "Lorena" is part of the score in "The Searchers" as stated above. "I'll Take You Home Again, Kathleen" was used several times by Ford. I believe it is "The Isle of Innisfree" that is sung quite well by Maureen O'Hara in "The Quiet Man". The song I am looking for is the one called "Mush-Mush-Mush Tural-i-addy" sung by Barry Fitzgerald and others as they deliver the furniture the morning after the wedding. I'm sure this is traditional Irish and likely known by other titles with words that vary. Not only did Ford get a great feel for the time and place in his films by using folk tunes, he saved money by avoiding royalties to composers, and Ford generally brought in his films on time and under budget.