Product of the ESV Kaufbeuren youth system, Sulzer made his debuts in the Oberliga in 2000. A transfer in 2002 to the Hamburg Freezers allowed him to play his first few games in the DEL in 2002-03. His good play earned him to be drafted to the NHL by the Nashville Predators in the 2003 NHL Entry Draft, 92nd overall.
The following season, he joined DEG. Sulzer developped into a reliable starter over time, which resulted in having the chance to be a member of the German National Team. His excellent play resulted in the IIHF Directorate Award for the Best Defender of the 2004 World Junior Championship Division I, Groupe A.
On June 1st 2007, he signed a two-year contract with the Predators and has played the 2007-08 season with the Milwaukee Admirals. He earned his first game in the NHL on January 15th 2009 against the Montreal Canadiens, following a first call-up with the big club a week earlier during which he did not play. At the time of his NHL debuts, Sulzer 29 points in 39 games in the AHL - he was third among the defencemen in the league.