Rinne began his career in 1975, playing for Lukko Rauma of the SM-liiga, where he strongly impressed, notching 36 goals and 22 assists in 36 games, thus finishing as the league's top goalscorer and earning himself a spot on the league's all-star team. His strong debuts impressed the north american scouts - he was drafted to the WHA in the 9th round, 101st overal by the Phoenix Roadrunners in the 1976 WHA Amateur Draft, and by the California Seals, in the 6th round, 95th overall, in the 1976 NHL Entry Draft.
Rinne, however, never played in North America. He rather stayed with Rauma until 1980, where, for the season 1980-81, he played for the HPK Hämeenlinna. He would return to Rauma the next season, before heading in 1982 for the FPS Forssa. Two seasons later, he went to KalPa Kuopio, whom he helped promote to the SM-liiga in 1986. He spent 6 successful seasons with Kuopio before leaving the club to join KooKoo Kouvala for 1990-91, after which he retired. He however returned to play in 1997-98 with YJK Ylivieska of the II.Divisioona. The next season, he played one game with KalPa, then 3 the year after with the JHT Kalajoki before retiring for good from the competition.
He established the SM-liiga's record for the quickest two goals on January 16th 1977, scoring two goals 6 seconds apart.
In addition to his league career, Rinne took part in the 1976 Canada Cup, the 1976 World Championships and the 1979 World Championships. His number 27 was retired by KalPa; he was also inducted to the Finnish Hockey Hall of Fame, where he is Jääkiekkoleijona # 168.