Byrne was a great ambassador for American chess, seemingly on good terms with players from both sides of the Iron Curtain. At the 1966 Chess Olympiad in Havana, Cuba, Bobby Fischer, a member of the Worldwide Church of God, would not compete on Saturdays, and the tournament officials knew this, yet they scheduled his first game against a Soviet player on Saturday, leading to accusations and hot tempers by the U.S. and Soviet teams and the tournament officials. Byrne's diplomacy and communication skills and the respect that all the players had for his integrity were enough to get the game rescheduled with everyone saving face. The tournament proceeded without further incident. Host Fidel Castro gave Byrne a beautifully hand-carved chess set to thank him.