Grenadier's father, Albert, was a trumpet player, and his two brothers, Phil and Steve, play trumpet and guitar, respectively. Grenadier began on trumpet when he was 10 years old before beginning to play the bass the following year. Grenadier's father helped introduce him to the instruments and music theory. Larry's older brother Phil began listening to jazz around this time, influencing his sibling's musical interests. Grenadier began listening to several jazz bassists including Ray Brown, Charles Mingus, Richard Davis, Paul Chambers, Wilbur Ware and Oscar Pettiford, among others.