A graduate of the WHL's Red Deer Rebels, Asham is known for his toughness rather than his finess or scoring ability. He averaged more than 150 penalty minutes in junior, but just as impressive was his ability to score--88 goals in his final two years.
As a result, Asham was drafted by Montreal 71st overall in 1996 NHL Entry Draft. In his first season he saw limited action, and just when he thought he had made the team in 1999-00 he suffered an abdominal injury and missed half the season.
He made a bit of a recovery in 2000-01, but played only a little more than half a year. Throughout his first four seasons of professional hockey, Asham spent most of his time honing his skills in the AHL. In 2002-03 he got a fresh start to his career, when Montreal dealt his rights to the New York Islanders. Upon his arrival on Long Island, Asham became a one of the club's more durable and feisty players. After four years on the Island, Asham was signed by the Islanders' division foes in New Jersey on August 7, 2007.
Asham appeared in 77 games with the Devils and his crash and bang style was welcomed on a relatively soft 2007-08 New Jersey squad. However, after the Devils were eliminated in the first round of the NHL playoffs, Asham's stint in New Jersey came to an abrupt end. An unrestricted free agent, Asham signed in the summer of 2008 with yet another Atlantic Division club, the Philadelphia Flyers.
Following a Cinderella-type run to the Stanley Cup Final with the Flyers in 2009-10, Asham found himself again an unrestricted free agent and on the move once more. He again chose to stay within the Atlantic Division, inking a one-year deal with the Pittsburgh Penguins.