János Urányi worked as a laborer in Csepel and learned about kayaking during a vacation. At the 1948 London Olympics he finished fifth in the 10,000 meter kayak doubles along with Gyula Andrási. In the 1000 meters, paired with Kálmán Blahó, he was disqualified due to leaving the lane in the final. In Helsinki he started in the individual 1000 meters, but was eliminated in the qualifiers. In 1952 he moved to the Budapest Dózsa. At the 1954 World Championships he finished in first place in a two-person kayak, but the jury placed them in third place due to the obstruction of the Swedish team. At the 1956 Melbourne Olympics he became Olympic champion in the 10,000-meter kayak doubles along with László Fábián. This victory was the first Olympic gold medal for Hungarian kayaking.
At the 1957 European Championships he won a gold medal in the 500-meter kayak and a silver medal in the 10,000-meter kayak. At the 1958 World Championships, he won his only world championship title at 10,000 meters, along with László Fábián. He was a bronze medalist at the European Championships held a year later, before retiring two years later. During his career he won the national championship twenty-five times. From 1961 he worked in Újpesti Dózsa where he became the head of the shipbuilder’s workshop. He died in 1964 at the age of 39 due to illness.