A native of Ylitornio, Finland, Koivisto moved to Rauma as a teenager in 1998 to play for Lukko, as northern Finland lacked an SM-liiga team at that time. After one year spent with the Lukko junior A team, Koivisto debuted in the SM-liiga in the 1999–2000 season, although he still split his time between the adult and the junior teams. In 2000, Koivisto appeared in the U18 World Championships and won the gold medal with Team Finland. He also represented Finland in the 2001 and 2002 World Junior Championships, coming home with silver and bronze, respectively.