Kaunismäki began playing in Finland with the Jokerit Helsinki organization's junior teams. The later played for Kiekko-Vantaa of the third-level Suomi-sarja and the second-level Mestis. Kaunismäki played a total of five seasons in Mestis league, two with Kiekko-Vantaa and three with Ahmat Hyvinkää. In 2003 he joined Stavanger of the Norwegian league.
Kaunismäki was selected to play for the Norway men's national ice hockey team at the 2010 Winter Olympics. He previously represented Norway at 2008 and 2009 World Championships after changing his citizenship.