Woody confirmed his move with a tweet, thanking the Jets and their fans.
The moves will help New York navigate the salary cap once a new collective bargaining agreement is in place. Contracts will expire for 17 Jets this week. Among that group the Jets want to re-sign are receivers Braylon Edwards, Santonio Holmes and Brad Smith, and cornerback Antonio Cromartie.
Each player the Jets are about to cut was significant in his own way. Three of them combined for 11 Pro Bowls and five first-team All-Pro selections. The other will go down as one of the biggest draft busts in NFL history.
Woody started for one of the NFL's best offensive lines, but he also missed the final three regular-season games to rest his right knee and the final two playoff games because of an Achilles injury that required surgery.
Taylor was a strong veteran presence. The six-time Pro Bowler, 2006 Associated Press Defensive Player of the Year and 2007 Walter Payton Man of the Year recorded five sacks, two forced fumbles and two recoveries.
Jenkins played only one game. A left knee injury ended his 2009 campaign after six games. He returned only to suffer a repeat injury six plays into the season opener.
Gholston was the sixth overall pick of the 2008 draft. He was supposed to be a monstrous pass-rusher and entered the league with a contract that included $21 million in guarantees.
But Gholston didn't record a single sack in his three seasons. He started only five games as a Jet, and occasionally was a healthy scratch.