Chosen by the Dallas Stars with the No. 29 pick in the 2013 NHL Draft, Dickinson became an NHL regular with Dallas during the 2018-19 season, finishing with 22 points (six goals, 16 assists) in 67 games. He followed that with 21 points (nine goals, 12 assists) in 65 games in 2019-20, then scored four points (two goals, two assists) in 27 postseason games, helping the Stars reach the Stanley Cup Final for the first time since 2000 before losing to the Tampa Bay Lightning.

The center, a native of Georgetown, Ontario, played four seasons of junior hockey for Guelph of the Ontario Hockey League. During that time, he was named to the OHL Second All-Rookie Team, competed in the 2013 CHL Top Prospects Game and won gold for Canada at the 2013 IIHF World Under-18 Championship.

In 2013-14, Dickinson had 78 points (26 goals, 52 assists) to help Guelph win the OHL championship before losing to Edmonton in the Memorial Cup final. He signed a three-year, entry-level contract with the Stars on May 29, 2014, and scored 71 points, including 27 goals, for Guelph in 2014-15.

After scoring 22 goals for Texas of the American Hockey League in 2015-16, he scored in his NHL debut April 7, 2016. Dickinson bounced between Dallas and Texas each of the next two seasons and helped the AHL affiliate reach the Calder Cup Final in 2018.