Acy attended John Horn High School in Mesquite, Texas. As a senior in 2007–08, he averaged 17.8 points and 7.8 rebounds per game. Considered a four-star recruit by Rivals.com, Acy was listed as the No. 25 power forward and the No. 84 player in the nation in 2008.
As a freshman at Baylor in 2008–09, Acy earned Big 12 All-Freshman team honors from the media and was named Big 12 Co-Rookie of the Week on November 24, 2008. He ranked sixth on team in scoring (5.4) and fourth in rebounding (3.6), while leading Bears in blocked shots (34). He set Big 12 and Baylor records with 20 consecutive field goals made—he made his first 20 field goals of his career, starting on November 15 and ending on November 24.
As a sophomore in 2009–10, Acy averaged 9.3 points and 5.1 rebounds per game as Baylor's sixth man. He was subsequently named to the Big 12 All-Reserve team by the media.
As a junior in 2010–11, Acy averaged 12.4 points and 7.6 rebounds per game. In the first round of the Big 12 Men's Basketball Tournament, Acy recorded a team-high 21 points and a career-high 15 rebounds in a loss to Oklahoma. Following the season, he earned the Big 12 Sixth Man Award and was named All-Big 12 Honorable Mention.
As a senior in 2011–12, Acy averaged 12.0 points and 7.4 rebounds per game. Following the season, he was named to the All-Big 12 second team and the Big 12 All-Defensive team.
On June 29, 2012, Acy was selected by the Toronto Raptors with the 37th overall pick in the 2012 NBA draft. On July 16, 2012, he signed a multi-year contract with the Raptors. He played sparingly for Toronto in 2012–13 and earned two assignments to the NBA Development League, where he played for the Bakersfield Jam. On April 6, 2013, Acy scored a season-high 13 points in the Raptors' 100–83 loss to the Milwaukee Bucks.
On December 9, 2013, Acy was traded, along Rudy Gay and Aaron Gray, to the Sacramento Kings in exchange for Greivis Vásquez, Patrick Patterson, John Salmons and Chuck Hayes. Six days later, he made his debut for the Kings in a 106–91 win over the Houston Rockets, recording four points, three rebounds and one block in 13 minutes off the bench.
On August 6, 2014, Acy was traded, along with Travis Outlaw, to the New York Knicks in exchange for Wayne Ellington and Jeremy Tyler. He made his debut for the Knicks in their season opener on October 29, 2014, recording six points, six rebounds, one assist and one block in 21 minutes off the bench in a 104–80 loss to the Chicago Bulls. On January 5, 2015, he recorded career highs of 19 points and 14 rebounds in a 105–83 loss to the Memphis Grizzlies. He had a career-best season in 2014–15, averaging career highs in points (5.9), rebounds (4.4), assists (1.0), minutes (18.9) and games played (68).
On July 22, 2015, Acy signed with the Sacramento Kings, returning to the franchise for a second stint. On January 7, 2016, he scored a season-high 18 points in a 118–115 win over the Los Angeles Lakers.
On July 20, 2016, Acy signed with the Dallas Mavericks. On November 18, 2016, he was waived by the Mavericks after appearing in six games. On November 28, 2016, he was acquired by the Los Angeles D-Fenders of the NBA Development League and then immediately traded to the Texas Legends, the Mavericks' D-League affiliate. On December 1, 2016, he made his debut for the Legends in a 121–106 win over the Greensboro Swarm, recording 16 points, seven rebounds and two blocks in 17 minutes as a starter. In 12 games for the Legends, he averaged 17.3 points, 8.1 rebounds, 1.2 assists, 1.6 steals and 2.1 blocks in 31.9 minutes.
On January 10, 2017, Acy signed a 10-day contract with the Brooklyn Nets. He made his debut for the Nets that night in a 117–97 loss to the Atlanta Hawks, playing the final 1:17 and scoring four points. On January 20, he signed a second 10-day contract with the Nets. The following day, he had a season-high 14 points off the bench in a 112–105 loss to the Charlotte Hornets. On January 30, 2017, he signed a multi-year contract with the Nets.