He played college basketball in the United States for the University of Kansas for four seasons, where he won an NCAA championship in 2008 before being selected with the 56th overall pick by the Seattle SuperSonics in the 2008 NBA draft.

He graduated with a degree in computer science and was selected to the Academic All Big 12 Basketball Team three times, in 2006, 2007, and 2008.

[7] In his "contract year", Kaun had the best season since coming in the club, starting in all 30 games played in the Euroleague and averaging a career-high 9.9 points and 4.5 rebounds on 69.1% shooting from the field.

After winning his fifth VTB United League championship with CSKA, there were some reports that Kaun had retired from professional basketball.

[11] He made his debut for the Cavaliers on October 28, recording one rebound, one assist, one steal and one block in a 106–76 win over the Memphis Grizzlies.

[15] He was reassigned to Canton the next day to help the team in Game 1 of the D-League Eastern Conference Finals against the Sioux Falls Skyforce.

On July 15, 2016, Kaun was traded, along with cash considerations, to the Philadelphia 76ers in exchange for the rights to Chukwudiebere Maduabum.