Graham was recruited by Geelong at pick 40 in the 1992 Australian Football League (AFL) pre-season draft.

Graham was noted for his long kicking (in particular his expertise at the torpedo punt),[5] and could play at both ends of the ground as a key-position player.

In 2001, he was inducted into the Geelong Football Club Hall of Fame and granted life membership.

Individual Graham was first approached by the New York Jets to try out as a punter in 1997; however, he decided to stay in the AFL.

[7] In 2004, he surprised many in Australia when he quit Australian rules football and left for the United States to pursue a career as a punter in the National Football League, where he would ultimately be paid many times more than he would have earned in Australia and extended his playing career by many years.

[7] He moved to New York City and attended Jets training camp at Hofstra University, before being confirmed as the team's starting punter for the 2005 NFL season when his predecessor Micah Knorr was released.

[5] In late 2006, Graham made history becoming the first Australian to captain an American pro sports team when he joined the leadership group of the New York Jets.

The team had released incumbent punter Steve Weatherford earlier in the week due to travel arrangement issues.

In the NFC Championship game that year, Graham's Cardinals played the Philadelphia Eagles, whose punter, Sav Rocca, also was an ex-AFL player from Collingwood and North Melbourne.

In 2011 with the NFL lockout looming, Graham announced his intention to play for the Australian National team in the 2011 IFAF World Cup, the official international American Football championship tournament.

Graham's father and his younger brother, Sam, attended Super Bowl XLIII in Tampa, Florida, to watch him when he played for the Arizona Cardinals.