At the age of 18, Richárd Bicskey became a member of the national team and also won his first national championship title, which was followed by another 32 national titles during his career, mainly in track cycling. Due to the revolution, he missed the 1956 Melbourne Olympics, but with Imre Furmen (his tandem partner), went to compete in Paris, where Bicskey became the French amateur sprint champion. They returned in 1957 and resumed racing in Hungary. In 1958, as a member of the Hungarian team, Bicskey took second at the European Championships in the team pursuit, and in 1964 he finished fifth in tandem at the Tokyo Olympics with Ferenc Habony. That year he also placed 10th in team pursuit at the World Championships. In 1967 he finished fifth in tandem at the World Championships, by then alongside László Sleisz. Bicskey retired in 1968 and started working as a coach. He lived in the Netherlands for three decades for family reasons, returning home in the mid-2000s.