Folk, 32, who joined the Jets in 2010, is the second-leading scorer in franchise history (729 points), behind only Pat Leahy. His 81.3 percent accuracy on field goals is the highest in team history based on a minimum of 100 attempts.
"It's a sad day, but that's the business side of things," Folk said in a statement. "I had a great seven years here. I think the only thing that would've topped it off would've been a couple of Super Bowl wins."
Folk, best remembered for a game-winning kick in the 2010 wild-card game against the Indianapolis Colts, is coming off one of his best years statistically. He made 87 percent of his field goals (27-for-31). He missed two extra points, including a costly miss in the opener.
The Jets have no experienced kickers on the current roster. Folk, who had one year left on his contract, had a $3.6 million cap charge. The Jets will recoup $3 million in cap room.
Giacomini, 31, who signed a free-agent deal with the Jets in 2014, played only five games last season due to a back problem. He underwent surgery in January to repair a disc, saying he'd be ready by training camp.
He, too, had only one year left on his contract. He was due to count $5.1 million on the cap, but $4.5 million of that will come off the books.
The Jets could replace Giacomini by handing the job to second-year tackle Brandon Shell, who started the final three games and received positive reviews.