He played for HC Dukla Trenčín of the Slovak Extraliga from 1999 to 2004. The Toronto Maple Leafs chose him 113th in the 2004 NHL Entry Draft. Kukumberg played in 2004-05 with the Neftekhimik Nizhnekamsk in Russia before signing a two-way contract with the Leafs in 2005. He played all of 2005-06 with the Toronto Marlies of the American Hockey League.
Kukumberg returned to Europe in 2006 and remains there.