He has previously played for the Boston Bruins, Florida Panthers, St. Louis Blues, and Toronto Maple Leafs.

[1] Acciari attended Bishop Hendricken High School in Warwick, Rhode Island, where he won the state championship in his freshman year in 2007 and was named team captain in 2009.

[6] An undrafted free agent, on June 8, 2015, the Boston Bruins of the National Hockey League (NHL) signed Acciari to a two-year entry-level contract.

[9] He registered his first NHL point on March 7, assisting on Brett Connolly's first period goal in a 5–4 victory over the Florida Panthers.

[12] Acciari appeared in three playoff games for Providence, scoring one goal and three points as they were eliminated by the Wilkes-Barre/Scranton Penguins in the opening round.

[26] He returned to the lineup on November 10[27] and helped solidify the fourth line with Sean Kuraly and Tim Schaller.

[29] He appeared in 12 playoff games, scoring the one goal and two points as the Bruins were eliminated by the Tampa Bay Lightning in the second round.

[33] As an unrestricted free agent, Acciari signed a three-year, $5 million contract with the Florida Panthers on July 1, 2019.

[39] Playing on a line with Jonathan Huberdeau and Vincent Trocheck, he scored the first two goals on Ben Bishop who was pulled by the Stars and replaced with Anton Khudobin.

[43] Acciari appeared in four playoff games with the Panthers, going scoreless, which were eliminated by the New York Islanders in the opening qualifying round.

[45] The Panthers made the 2021 Stanley Cup playoffs and he appeared in six games, going scoreless, as the team was eliminated in the first round by the Tampa Bay Lightning.

[46] He began the 2021–22 season on injured reserve after suffering a pectoral injury in a preseason exhibition game against the Lightning on October 5.

[47] The injury required surgery and after recovering, he was assigned to the Panthers' AHL affiliate Charlotte Checkers for a two-game conditioning stint.

[49] He played in 20 games with the Panthers, scoring three goals and eight points as the team won the Presidents' Trophy for the first time in franchise history as the best squad in the league.

[50] In the 2022 Stanley Cup playoffs, he played in nine games, going scoreless, as the Panthers were upset in the second round by the Tampa Bay Lightning.

[58] Acciari finished with four goals and five points in 23 games with the Maple Leafs during the regular season and was one of Toronto's key penalty killers after arriving.

[60] 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.

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

[63] He registered his first point for Pittsburgh on November 4 assisting on Matt Nieto's first goal of the season in a 10–2 drubbing of the San Jose Sharks.