Born in Cork, Ireland, Dunne is the nephew of former Irish Prime Minister, Jack Lynch. In 1974 he graduated from University College Cork and Montfort College of Performing Arts after studying music. That same year he won the Castlebar Song Contest with the song "Shalom". Two years later he represented Ireland at the Yamaha Music Festival with his own composition "Lover, Not a Wife". Also in 1976 he released a single, "Bad Boy" as well as his first top ten hit in Ireland, "Danny". During these years he played with his backing band, Stateside (and later, The Formula), on the Irish club circuit.
In 1979 he entered his own composition "Happy Man" into the Irish National Song Contest. Competing against former Irish Eurovision entrants Tina Reynolds and Red Hurley and future winner Johnny Logan, he won the contest easily with 36 points and thus gained the right to represent Ireland at that year's Eurovision Song Contest. Performing in Jerusalem at the final in March, he achieved a respectable fifth place with 80 points. The single went on to sell well in Europe and reached No.3 in the Irish charts. The song was produced by Nicky Graham.
In February 1982 he played some concerts in Michigan in the United States. These tempted him to move there permanently and the following year, Dunne emigrated from Ireland. Since then, Dunne has made a name for himself as a singer of traditional Irish ballads and has released a number of albums. He has also performed comedy routines and show tunes.