He began his career with HC Litvínov of the Czechoslovak First Ice Hockey League; his 49 goals in 1989–90 was the second highest total in Czechoslovakian history.

Reichel played 11 National Hockey League (NHL) seasons for the Calgary Flames, New York Islanders, Phoenix Coyotes and Toronto Maple Leafs.

A two-time Olympian, Reichel scored the lone shootout goal to eliminate Canada en route to a gold medal victory for the Czech Republic at the 1998 Winter Olympics.

Reichel began his elite playing career at the age of 16 when he joined CHZ Litvínov of the Czechoslovak First Ice Hockey League.

[8] A labour dispute resulted in the cancellation of the first half of the 1994–95 NHL season, and Reichel spent the time with the Frankfurt Lions of the Deutsche Eishockey Liga (DEL).

[7] Reichel and the Flames resolved their impasse prior to the 1996–97 NHL season and agreed to a three-year, US$4.4 million contract.

[11] The Flames sent him to the New York Islanders on March 18, 1997, in exchange for Marty McInnis, Tyrone Garner and a sixth round selection in the 1997 NHL Entry Draft.

[2] Reichel once again returned to North America as the Coyotes traded his NHL playing rights to the Toronto Maple Leafs, along with Travis Green and Craig Mills, in exchange for Danny Markov on June 12, 2001.

He spent the final six seasons of his playing career with Litvínov and posted a peak offensive total of 47 points in 52 games in 2006–07.

[7][18] Having earned a spot with the senior team at the 1990 World Championships, Reichel was placed on a line with two other young players, Bobby Holík and Jaromír Jágr.

The line starred at the event; Reichel recorded 11 points in the tournament as Czechoslovakia won the bronze medal.

Reichel scored the lone goal and Dominik Hašek stopped all Canadian shots to lead the Czech Republic into the gold medal game.

Martin, who moved to Germany in 1990, represented his adopted nation at the 2002 Winter Olympics and played against Robert's Czech Republic.

[22] His son Kristian Reichel, also a hockey player, plays for the Winnipeg Jets affiliate Manitoba Moose.