Boston Bruins

On June 8, 2015, the Boston Bruins of the National Hockey League (NHL) signed Acciari to a two-year entry-level contract.^[5] In the midst of the 2015–16 season, on March 1, 2016, Acciari was called up to the Bruins for the first time as a professional. He made his NHL debut that night in a home game against the Calgary Flames.^[6] It would not be until March 28, 2017, that Acciari would score his first NHL goal in a home game against the Nashville Predators; Acciari scored the third Bruins goal en route to a 4–1 regulation-time Bruins win.^[7][8] On June 28, 2017, Acciari signed a two-year contract extension with Boston.^[9]

Florida Panthers

As an unrestricted free agent, Acciari signed a three-year, $5 million contract with the Florida Panthers on July 1, 2019.^[10] Following the signing, Panthers General Manager Dale Tallon praised Acciari's versatile play and physicality, saying: "This guy's going to make people accountable, and make our own players accountable as well. He's a versatile player. He will take some of the burden away from others by taking D-zone faceoffs, killing penalties and playing that physical role. He gives our fourth line a little more spirit and passion."^[11] However, he was unable to ever play a complete 82 regular-season with the Panthers due to the COVID-19 pandemic. In his first season, he tallied a career-best 20 goals and 27 points through 66 games.^[12]

St. Louis Blues and Toronto Maple Leafs

Following the conclusion of his contract with the Panthers, Acciari left as a free agent to sign a one-year, $1.25 million contract with the St. Louis Blues on July 13, 2022.^[13] He skated in 54 games for the Blues during the 2022–23 season, recording 10 goals and eight assists.

On February 17, 2023, Acciari and Ryan O'Reilly were traded by the Blues to the Toronto Maple Leafs in a three-team trade also involving the Minnesota Wild.^[14] Acciari finished with four goals in 23 games with the Maple Leafs during the regular season and was one of Toronto's key penalty killers after arriving.^[15] He scored the game winning goal in a 3–2 win over the New York Rangers in the Maple Leafs's final game of the season on April 13.^[16] During the first round of the 2023 Stanley Cup playoffs, Acciari broke his own 25-playoff game goalless drought when he scored the opening goal in Toronto's 4–3 win over the Tampa Bay Lightning.^[17] The Maple Leafs eliminated the Lightning, but were defeated by the Florida Panthers in the second round.^[18]

Pittsburgh Penguins

As a free agent from the Maple Leafs, Acciari was signed to a three-year, $6 million contract with the Pittsburgh Penguins on July 1, 2023.^[15]