Mednis finished second in the 1955 World Junior Championship behind Boris Spassky (the two drew their game). He was the first player to beat Bobby Fischer in a U.S. Championship. He played on the 1962 US team at the 15th Chess Olympiad and finished equal third in the 1961–62 U.S. Championship. Tournament results included third at Houston 1974, equal fourth at New York City 1980, and equal first at Puerto Rico 1984. The Puerto Rico Chess Federation, rather than the United States Chess Federation, formally proposed him for the Grandmaster title. He played in the 1979 Interzonal tournament in Riga, his birthplace.