Juan Rodríguez was interested in sports from a young age, but switched early from football to rowing. He had a successful international career after World War II, which began in earnest when he won a bronze medal in the double sculls event at the 1948 London Olympics, alongside William Jones. Rodríguez repeated this feat four years later in Helsinki, this time with Miguel Seijas. He also earned sculling titles at the 1954 and 1956 South American Championships, and captured silver in the single sculls at the 1955 Pan American Games. By career he worked for ANCAP, Uruguay’s national petroleum company. In 2019 the city of Paysandú named a coastal walkway in his honor.