Kennedy is the grandson of Celtic and Manchester United player Jimmy Delaney, but he is not related to John Kennedy, a midfielder who played for Celtic in the 1960s. He came through Celtic’s youth system and signed his first professional contract on the same day as fellow defender Stephen McManus. He made his first-team debut aged 16 during the 1999–00 season. In the 2003–04 season, he had a run in the Celtic starting eleven and came under scrutiny from scouts of AC Milan who contacted Celtic about making a move. It was during this season that Kennedy scored his only goal for Celtic, against Dundee in the SPL.
Kennedy is currently defensive coach under Brendan Rodgers.