Gramatica, a fourth-round draft pick in 2001, suffered significant injuries in two of his three seasons in Arizona.
He played just four games last season because of a back ailment that eventually required surgery. He was replaced by Neil Rackers, the only kicker currently under contract with the team.
"I'm not bitter or upset at the Cardinals," Gramatica told the Republic. "They treated me well. That's life, and that's how football goes."
Gramatica was due to become a restricted free agent March 2, and the team would have had to extend him a tender offer to retain his rights.
He is rehabilitating from surgery, he said, and hopes to resume kicking in about six weeks, according to the newspaper.
Gramatica missed the final three games of his rookie season after suffering a torn knee ligament while celebrating a field goal.
Rackers, who has played three years, made 9 of 12 field-goal tries and was perfect on eight point-after attempts.