The River Roe was sung to a variant of the Tramps and Hawkers/ Caroline of Edinburgh Town tune. Additional information can be seen at Andrew Kuntz's The Fiddler's Companion:
A midi of the tune will go to The Mudcat Midi Pages. For the time being, it can be heard via the South Riding Folk Network site:
The River Roe Midi made from notation in The Irish Song Book (A.P. Graves, revised edition, 1895), where it was set to William Allingham's The Winding Banks of Erne. It appears to be the version printed in Joyce's Old Irish Folk Music, and is played through twice to accommodate the length of the stanzas.