After playing two seasons in the Ontario Hockey League for the Belleville Bulls, Bičánek made his National Hockey League debut with the Senators in the 1994–95 season, appearing in six games.

After appearing sparingly for the Senators during the 1997–98 and 1998–99 seasons, Bičánek was traded to the Chicago Blackhawks on March 12, 1999, in exchange for a pick in the 1999 NHL Entry Draft.

The Blackhawks left him unprotected for the 2000 NHL Expansion Draft, allowing him to be selected by the Columbus Blue Jackets for their inaugural season.

During the 2004–05 NHL lockout, Bičánek returned to the Czech Republic to play for HC JME Znojemští Orli.

After five seasons in Znojmo he moved to HC Kometa Brno.