A Jewish Hungarian, Róna played for BKE Budapest in 1936. Three years later, the situation for him in a very anti-semitic Hungary forced him to leave the country. He exiled himself in Switzerland for a year, where he played for EHC Arosa. The next season, he moved to Finland, where he played for Kronohagens IF; he became Finnish champion that year with them. In 1941, Róna moved again, this time to Sweden, where he joined Karlbergs BK. After the war, he moved to France and played with the Paris Lions for 1949-50, becoming the first Hungarian player to play hockey in France.
László Róna represented the Hungarian National Team in four different tournaments, including the 1936 Olympics in Garmisch-Partenkirchen. Ironically, in these Olympics held in Nazi Germany, all the teams had to make the Hitler salute prior to the initial faceoff of every matches.
Róna also played tennis; the versatile Hungarian was Swedish International Tennis Champion. With his sports career over, he moved to Spain where he settled.