Try the Roud Folksong Index, which returns 12 results for songs collected in Co. Longford. The first couple are versions of songs collected widely elsewhere (Green Broom, Green Grow the Rushes, Let no man steal your thyme), but interestingly of those 12 results, 5 of them appear completely unique to Longford (or at least, there are no other records in the index).

Those songs are:
Lovely Mary O; Paddy Maguire; The Door it was Left Down; The Schoolmaster and the Labourer; and Innocent Darby Dane