Dickson enrolled at the University of Oregon and was redshirted his freshman year. He spent time with the defensive ends, but ended up as the tight end in the team’s depth chart behind Dante Rosario. He played 13 games as a defensive lineman in 2006.

In 2009, Dickson was selected to the preseason watch list for the Mackey Award, given out to the nation’s best tight end. On September 27, 2009, Dickson was named the national player of the week by the Walter Camp Football Foundation. He earned the honors after catching 11 passes for 148 yards and three touchdowns in an upset of the #6 California Golden Bears.

Dickson was regarded as one of the best tight ends available in the 2010 NFL Draft.

Dickson was drafted in the third round (70th overall) by the Baltimore Ravens. The pick was one of three that the Ravens acquired in a trade with the Denver Broncos on the first night of the draft on April 22. The Broncos received a first-round pick which was used to select Tim Tebow. Dickson signed a three-year contract on July 2, 2010.

As a backup to veteran Todd Heap, Dickson recorded 11 receptions, 153 yards, and one touchdown. However, the following year marked big changes for Dickson. Heap was released, and Dickson found himself in a battle with Dennis Pitta for the starting job. Also, Dickson switched from #83 to #84 after Lee Evans became a Raven. Dickson was the main starter in 2011, and caught five touchdowns.

During the Ravens Super Bowl championship season in 2012, Dickson saw his role in the passing game decrease in favor of Pitta, though he often found himself on blocking duty.

During the 2013 season, however, Dickson was the starter for most of the season as Pitta suffered a hip fracture injury before the season began.

On April 10, 2014, Dickson signed a one-year deal with the Carolina Panthers.

Dickson signed a three-year, $6.8 million contract extension on March 11, 2015. During the 2015 season, the Panthers won 15 games, which set a team record for victories.

On February 7, 2016, Dickson was part of the Panthers team that played in Super Bowl 50. In the game, the Panthers fell to the Denver Broncos by a score of 24–10.

On October 8, 2017, Dickson posted an impressive performance with 175 receiving yards as the Panthers won 27-24 over the Detroit Lions.

On March 16, 2018, Dickson signed a three-year deal with the Seattle Seahawks worth up to $14 million, with the first year fully guaranteed. He was placed on the reserve/non-football injury list on September 1, 2018. He was activated to the active roster on October 27, 2018.

On September 2, 2019, Dickson was placed on injured reserve. He was designated for return from injured reserve on October 30, 2019, and began practicing with the team again. He was activated on November 20, 2019. However, he was placed back on injured reserve two days later.

On March 31, 2020, Dickson was released by the Seahawks.