Born in 1947, Molloy is a native of county Roscommon, in the western area of Ireland. Molloy is widely recognized as the best flute player in Irish traditional music and was formerly in The Bothy Band and Planxty.

He started playing at the age of 12, and participated in traditional music festivals (called fleadhanna cheoil) where he won numerous competitions. In 1964, he moved to Dublin where he not only performed in clubs but also studied aeronautical engineering. He met Paddy Moloney at the Old Sheiling and soon the two were playing duets around town.

By the 1970s, Molloy was working as an airline mechanic and playing music on the side. When the traditional music scene boomed, he tried his hand at being a professional musician after securing a six-month leave of absence from his company. He never returned. He was a founding member of The Bothy Band, through which he became known as Irish music's premier flute player. Shortly after the group disbanded in 1979, Molloy joined Planxty, before becoming one of The Chieftains.