 Döme has played in Europe since 2003, playing two seasons with Södertälje SK of the Swedish Elite League and the 2005–06 season with the Nürnberg Ice Tigers of Germany's DEL.In July 2006 he signed a two-year contract with Modo Hockey of the Swedish Elite League where he became a Swedish champion in the season of 2006/2007.Received an award as the “Modo player with the largest heart for the spectators” during the celebrations at the town square the day after he brought the gold to Modo and Örnsköldsvik.