Cichocki began attracting attention while playing in the Michigan National Hockey League during high school. While playing for the Grosse Point Majors, Cichocki put up 50 goals and 70 assists in 70 games to lead the league with 120 points for the 1979–80 season. He then spent two years playing in the Great Lakes Junior Hockey League before deciding to attend college at Michigan Technological University in Houghton, Michigan. Beginning in 1982, Cichocki began to put up big numbers with his NCAA club, including 54 points in his senior season with the Huskies. The hockey world took notice of his skills and Cichocki was signed as a free agent by his hometown team, the Detroit Red Wings, on June 28, 1985.