Volek was born in the former Czechoslovakia, where he played in 1983–84 for HC Slavia Prague. He later moved to HC Sparta Praha and became a member of the Czecholovakian national ice hockey team for the 1988 Winter Olympics. In 1988 he gained permission from the Czech authorities to visit his parents in West Germany and used this permission to defect to Canada. He joined the New York Islanders, having been selected by them in the 1984 NHL Entry Draft, and was named to the NHL All-Rookie Team in 1988–89.