Morrow played major junior hockey in the Western Hockey League (WHL) for the Portland Winter Hawks. During his junior career, he helped Portland win the 1998 Memorial Cup during the 1997–98 season.
Internationally, Morrow represented Canada, first winning a silver medal at the 1999 World Junior Championships with the junior team. He has also represented the senior Team Canada squad, most often at the Ice Hockey World Championships (four times), winning a gold medal in 2004 and a silver medal in 2005. He also played for Canada at the 2004 World Cup of Hockey, which Canada won. Morrow was named to Canada's roster for the 2010 Winter Olympics, where he scored two goals and added an assist in the tournament as Canada won gold.
Morrow played minor hockey in his hometown of Carlyle, Saskatchewan, and in the nearby community of Estevan. Despite putting up impressive numbers in his final year of bantam hockey (117 goals, 72 assists in 60 games), Morrow was not selected in the WHL Bantam Draft, and instead walked-on with the Portland Winter Hawks.
Morrow spent his entire junior career with the Winterhawks. In his rookie WHL season, he played in 65 games and recorded 25 points (13 goals and 12 assists). Later in his junior career, Morrow contributed more offensively and in his final three seasons in the WHL, he scored 88, 86 and 85 points, respectively. After the 1996–97 season, Morrow was selected by the Dallas Stars in the first round, 25th overall, of the 1997 NHL Entry Draft. During the 1997–98 WHL Season, Morrow helped the Winterhawks capture the Memorial Cup. During the 1998 Memorial Cup tournament, Morrow scored one goal and added two assists in four games. He represented the Western Conference in the WHL's All-Star Classic three times, in 1997, 1998 and 1999. After the 1998–99 season, Morrow was named to the WHL's Western Conference First All-Star Team and also to the CHL Third All-Star Team.
Morrow began his professional career in 1999. He split the 1999–2000 season between the NHL's Dallas Stars and the Michigan K-Wings of the International Hockey League (IHL). He played his first game with the Stars on November 18, 1999, against the Philadelphia Flyers, and recorded his first points (a goal and an assist) on November 22, 1999, against the Colorado Avalanche. After this season, Morrow was named the team's Rookie of the Year.
After his rookie campaign with the Stars, Morrow became a regular fixture in the Dallas lineup, playing at least 70 games in each season until the 2006–07 season, when he missed 33 games after suffering severed tendons in his wrist. He bounced back with a strong 2007–08 season, where he played in all of the Stars' 82 games, but a knee injury (an ACL tear) limited him to just 18 games in 2008–09. He has often played through injuries, including a return to play despite breaking a bone in his ankle during the 2000 Stanley Cup playoffs.
Morrow played in the NHL YoungStars Game at the 2002 NHL All-Star Game, which was held in Los Angeles, California, where he scored a goal and added an assist. During the 2001–02 NHL season, he recorded his 100th career NHL point against the San Jose Sharks. He recorded his 300th career NHL point against the Chicago Blackhawks during the 2006–07 NHL season.
During the NHL lockout of 2004–05, Morrow signed a contract with the Oklahoma City Blazers of the Central Hockey League and played in 19 games.
Prior to the 2006–07 NHL season, Morrow was named the captain of the Stars, after the team asked incumbent captain Mike Modano to step down from the position. The Stars wanted to reward Morrow for his commitment to the team, after he bypassed free agency to sign a long-term (6 year) contract with the team. The Stars also felt the need to shift some leadership responsibilities to younger members of the team. Morrow's teammates and coaches greatly admired his dedication to the team, and believed he was an ideal choice to fill the role.
Heading into the 2009–10 season, Morrow ranked 14th on the all-time scoring list for the Stars' franchise. He has a knack for scoring big goals, as evidenced by his scoring three of the team's last four overtime winning goals in the playoffs.
Morrow was injured often during the 2011–12 season. He missed 25 games due to shoulder and upper back injuries, including 18 matches at the beginning of the 2012 year. Morrow returned to action on March 9 after the Stars activated him from injure reserve.
On March 24, 2013, Morrow was traded to the Pittsburgh Penguins, along with Dallas' third-round pick in 2013, in exchange for Joe Morrow (no relation) and the Penguins' 2013 fifth-round draft pick.
On September 23, 2013, Morrow signed a one-year deal as an unrestricted free with the St. Louis Blues.
On July 11, 2014, Morrow signed as a free agent to a one-year deal with the Tampa Bay Lightning. On June 17, 2015, during exit interviews, Morrow stated that he may "give it one more go," though he stated that he could change his mind in two weeks when it was time to resume workouts. Morrow stated that he had not spoken to Lightning General Manager Steve Yzerman about returning to the team, but he stated that he would love to return. Morrow said that he "had more fun here in this short time than I've had in I don't know how many years."
On March 17, 2016, Morrow announced his retirement. He had book-ended his career with Stanley Cup appearances in 2000 (with the Stars) and 2015 (with the Lightning).
Morrow's first international experience with Hockey Canada was at the 1995 World U17 Hockey Challenge in Moncton, New Brunswick, where he played for Team West and finished fourth. He was selected to Canada's team at the 1999 World Junior Ice Hockey Championships, which were held in Winnipeg, Manitoba. During the tournament, Morrow scored one goal and recorded seven assists in seven games. During this tournament, Morrow set a Canadian World Junior record with five assists in one game against Kazakhstan. Canada finished the tournament with the silver medal after losing to Russia in overtime in the gold medal game.
Morrow has also represented Canada at the senior men's level, playing in the World Championships four times (2001, 2002, 2004 and 2005) after his NHL season was over. In these four tournaments, he did not score a goal for Canada, but contributed with five assists. In 2004, Morrow helped Canada to a gold medal and in 2005, captured the silver medal. Morrow played in a single game for Canada at the 2004 World Cup of Hockey, where Canada captured the championship.
Morrow was invited to Canada's summer evaluation camp prior to the 2006 Winter Olympics in Turin, Italy, but was not selected for the final squad. In August 2009, Morrow participated in Hockey Canada's summer orientation camp prior to the 2010 Winter Olympics in Vancouver, British Columbia, Canada. After a strong start to the 2009–10 season, Morrow was ultimately selected as a member of Canada's Olympic Men's Hockey Team. He was selected to the team to provide toughness and grit. As the tournament progressed, Morrow found himself playing a larger role on a physical line with Ryan Getzlaf and Corey Perry of the Anaheim Ducks. Morrow finished the tournament with two goals and one assist, as Canada captured the gold medal with an overtime victory against the United States.
Morrow is the son-in-law of former Montreal Canadiens coach and former Dallas Stars teammate Guy Carbonneau, having married Carbonneau's daughter Anne-Marie on July 20, 2002. They have three children, a daughter, Bryelle, and twins; son Brody, and daughter Mallory, born May 11, 2008.Named to WHL West First All-Star Team (1999)
2004 Men's World Ice Hockey Championships gold medal
Gold medal in the 2010 Winter Olympics (Team Canada)