Throughout his career, he has accomplished various titles and achievements. Notably, he won the Cuban chess championship five times - in 1976, 1978, 1979, 1989 and 1990. Additionally, he won the Capablanca Memorial tournament in 2001.
Rodriguez Cespedes' impact also extends to the international stage; he represented his country in various chess Olympiads, starting from 1978 to 1992. His most significant achievement was a gold medal performance at the 30th Chess Olympiad in 1992, held in the Philippines.
He also played for Cuba in the 2005 World Team Chess Championship in Israel. Throughout his active years, Rodriguez Cespedes competed in several other tournaments and earned several prizes.