Lumme began his professional career in with Ilves Tampere of the SM-liiga in 1985–86. Following his rookie season in Finland, Lumme was selected by the Montreal Canadiens in the 3rd round (57th overall) of the 1986 NHL Entry Draft. He remained with Ilves Tampere for two seasons following his draft. In his final year before transferring to the NHL, Lumme scored 8 goals and 30 points over 43 games to help Ilves to the best regular season record in the league.