Bourne was drafted 38th overall by the Kansas City Scouts in the 1974 NHL Amateur Draft, though he would never play for the organization. On September 13, 1974, he was traded to the New York Islanders. Bourne spent the entire 1974–75 season with the Islanders but found himself in the minor leagues the following year. He returned to the Islanders for the 1976–77 season, and spent the next ten seasons on Long Island, recording at least thirty goals three times and between twenty and thirty goals three others. He won the Stanley Cup 4 times — 1980, 1981, 1982, 1983. Bourne's career ended on a high note. He was claimed by the Los Angeles Kings on waivers in October 1986 and played 2 seasons with the Kings. After retiring as a player, Bourne served as a coach for several minor league teams, including the Central Texas Stampede, Las Vegas Thunder and Utah Grizzlies. He now hosts the Hockey Greats Fantasy Camp in Kelowna, British Columbia.