Arizona also placed veteran offensive lineman Floyd Womack and rookie wide receiver Sean Jeffcoat on injured reserve due to shoulder problems.
Graham had served as the Cardinals' punter since December of the 2008 season and was a member of Arizona's NFC Championship team that year. The native Australian put together a banner 2009 campaign, averaging a career-best 47.0 yards per punt and placing an NFL-leading 42 kicks inside the 20-yard line, but slipped to a 43.4 average last season and was beaten out by ex-Brown Dave Zastudil this summer.
Zastudil, a nine-year veteran who was signed by the Cardinals on August 24, secured the job by punting twice for 95 yards in Thursday's preseason finale against Denver. The 32-year-old hasn't appeared in an NFL regular-season game since midway through the 2009 season, having spent all of last year on Cleveland's IR list with a knee injury.
Womack owns 71 career starts over 10 seasons with Seattle and Cleveland and was signed to a two-year deal by Arizona just prior to camp to provide experienced depth to the interior offensive line.
Other notable cuts made by the Cardinals included safety Matt Ware, defensive end Kenny Iwebema, linebacker Will Davis, fullback Reagan Maui'a and quarterback Brodie Croyle.
Ware had been with Arizona since 2006 and served as a core special-teams player for the club over that span. Davis, a sixth-round pick in 2009, started three games at outside linebacker last season and compiled 14 tackles and one sack. Iwebema was taken by the Cardinals in the fourth round of the 2008 draft, but recorded just 13 tackles and two sacks in three years.
Croyle, who made 10 starts over five seasons with Kansas City in 2010, was brought in by Arizona on August 24 but failed to pass returnee Richard Bartel for the No. 3 spot on the depth chart. The 28-year-old completed 10-of-16 passes for 125 yards in the Cardinals' 26-7 win over Denver on Thursday.
Maui'a, Arizona's primary fullback last season, was deemed expendable when the organization took rookie Anthony Sherman in the fifth round of this past April's draft.
Also released were running back William Powell, wide receivers Aaron Nichols and Isaiah Williams, tight end Steve Skelton, centers Ben Claxton and Kris O'Dowd, guard Tom Pestock, offensive tackles Cliff Louis and D.J. Young, defensive linemen Jeremy Navarre and Ricky Lumpkin, linebackers Cyril Obiozor, Pago Togafau and Kendall Smith, cornerbacks Fred Bennett, Marshay Green, Bryant Nnabuife and Thad Turner and safety Jared Campbell.
Ware's release could be a sign that Pro Bowl safety Adrian Wilson, who's missed the entire preseason with a partially torn biceps tendon, is on track to play in the team's September 11 opener against Carolina. The Cardinals are currently carrying only eight defensive backs on the active roster, along with five tight ends.