Rychel was originally selected 21st overall in the 2011 OHL Priority Selection by the Barrie Colts. Rychel was rated as a top prospect and was a first-round selection at the 2013 NHL Entry Draft. In his debut professional season in 2014–15, on an injury-plagued Blue Jackets roster, Rychel was recalled from AHL affiliate the Springfield Falcons and made his NHL debut on November 29, 2014, against the Nashville Predators. Rychel gained notoriety during the 2015–16 NHL season when reports surfaced that he had requested a trade out of Columbus over his lack of playing time at the NHL level. Rychel and his father, who also serves as his agent, held a meeting with management during the Blue Jackets development camp, raising concerns over Rychel being "buried" on the team's depth chart. Ultimately though, Rychel was not traded during the season, and split time between the club and their new AHL affiliate, the Lake Erie Monsters. The Monsters finished as one of the top teams in the league, and ultimately won the Calder Cup. On June 25, 2016, less than an hour after the conclusion of the 2016 NHL Entry Draft, Rychel was traded by the Blue Jackets to the Toronto Maple Leafs in exchange for defenseman Scott Harrington and a conditional 5th round pick. The Maple Leafs were rumoured to be one of the teams most interested in Rychel following his earlier trade request. On February 25, 2018, Rychel was traded along with Rinat Valiev and a 2018 second-round pick to the Montreal Canadiens in exchange for Tomáš Plekanec and Kyle Baun. As an unsigned restricted free agent with the Canadiens over the summer, Rychel was traded by the Canadiens to the Calgary Flames in exchange for fellow restricted free agent, Hunter Shinkaruk, on August 20, 2018, and signed a one-year contract with the Flames. On June 25, 2019, Rychel was not tendered a qualifying offer by the Flames, enabling him to become a free agent. Rychel paused his North American career, agreeing to his first contract abroad in signing a one-year contract with Swedish club Örebro HK of the Swedish Hockey League on July 27, 2019. Approaching the 2019–20 season, Rychel sought and was released from his contract with Örebro HK in order to later sign a one-year contract with Russian club HC Neftekhimik Nizhnekamsk on September 3, 2019. Rychel struggled to adapt to the KHL and was scoreless in just 7 games with Neftekhimik before his contract was terminated on October 8, 2019. Following his brief stint in the KHL, Rychel returned to North America and opted to resume his career in the AHL, by agreeing to a one-year contract with the Charlotte Checkers, affiliate of the Carolina Hurricanes, on November 19, 2019. He notched 1 assist in 6 games with the Checkers before he was suspended by the team and removed from the roster.
