Denney, 40, released a statement thanking and saying goodbye to Dolphins teammates, coaches and fans who have supported him since he joined the team in 2005. He has never missed a game in his 14-year career (224 games).
"John has done a tremendous job for this organization for a long time," coach Brian Flores said. "From a leadership standpoint, from a coming in everyday to work standpoint, no one has done more than he has. I'm very appreciative of the time I spent with him. At the end of the day, we felt like this was a move we needed to make."
The Dolphins signed former Giants long-snapper Taybor Pepper to take Denney's roster spot.
Releasing Denney was another Dolphins decision to get rid of a veteran before what is expected to be a rebuilding season.
The Dolphins have added 12 new players to their roster over the past four days. Miami has 31 players on the 53-man roster who are 25 or younger and just four who are 30 or older.
Miami's big move of the weekend was trading away left tackle Laremy Tunsil and receiver Kenny Stills for a bounty of picks. The Dolphins' roster underwent a complete remake this offseason under general manager Chris Grier and Flores.
"When you start something new, you never really know when you get into it. I didn't walk into it thinking we are going to do exactly this," Flores said. "You get in, you spend time with the team, you make assessments and evaluations, and you do what you feel is best at the end of the day. That's the process we've taken organizationally."
Miami also waived defensive tackle Vincent Taylor on Monday and signed former Giants defensive tackle John Jenkins to fill his roster spot.