Denney was born to Craig and Sheri Denney in Denver, Colorado. He learned to play the violin at age 5 and was an Eagle Scout. A member of the LDS Church, Denney served a church mission to Morristown, New Jersey. As a child, he was a fan of the Denver Broncos. Denney attended Westlake middle school. Denney was a three-year letterman at Horizon High School in Thornton, Colorado. As a senior, he earned first-team all-conference honors, second-team all-state honors and was his team's Special Teams Player of the Year. A three-time academic all-conference selection, Denney graduated from Horizon in 1997 with National Student-Athlete honors.
After a season at Ricks College, Denney transferred to Brigham Young University and was redshirted in 2001. A neck injury sidelined Denney early during his sophomore season in 2002, but he went on to start seven games at left defensive end for the team. He recorded 25 tackles on the season, including five-tackle performances against Colorado State and Utah. Additionally, Denney earned Academic All-Mountain West honors. As a junior in 2003, Denney earned Academic All-Mountain West honors for the second straight season while starting 12 games at defensive end. He finished the season with 42 tackles, six tackles for a loss and 4.5 sacks. He was named the team's Defensive Player of the Game twice, against USC and Wyoming. Denney recorded 27 tackles during his senior season in addition to 4.5 sacks and a forced fumble. Following the season, he was chosen to play in the Hula Bowl. During his three-year career at BYU, Denney started 29 of the 32 games in which he played and recorded 94 tackles. He also handled long snapping duties for the Cougars throughout his career. He majored in business management.
Professional career
Miami Dolphins
After going undrafted in the 2005 NFL Draft, Denney was signed by the Miami Dolphins. He beat out incumbent Ed Perry for the team's long snapping duties, holding the job the entire 2005 season and keeping it throughout the rest of his tenure. He was the only undrafted free agent for the Dolphins to appear in every game that season. He recorded five tackles on special teams during the year, including a season-high two stops against the San Diego Chargers on December 11. In February 2006, Denney underwent surgery on an injured elbow. He played in every game for the Dolphins for the second straight season, recording five tackles. Denney recorded two tackles and recovered a Brad Kassell fumble on a punt against the New York Jets on December 25, leading to a game-tying Olindo Mare field goal in the fourth quarter. On March 22, 2007, the Dolphins re-signed Denney to a four-year contract through the 2010 season. That season, he appeared in all 16 games for the Dolphins for the third straight year and recorded three special teams tackles. Through his first three NFL seasons, Dolphins placekickers made all 81 extra point attempts and 80.9 percent of the field goals for which Denney was the snapper. On August 15, 2013, Denney signed a three-year extension at an $1,150,029 average per year. On March 9, 2017, Denney re-signed with the Dolphins. On March 15, 2018, Denney re-signed with the Dolphins for one year. On February 8, 2019, Denney signed a one-year contract extension with the Dolphins. He was released on September 2, 2019 as the longest tenured Dolphin. in 2011.