The teacher, who has been named locally as Ronnie Smith, from Texas, was said to have been out jogging when the attack took place.
Adel al Mansouri, director of the school in Benghazi, said: "He was doing his morning exercise when gunmen just shot him. I don't know why. He was so sweet with everyone."
It was not immediately clear who was responsible for the killing.
Security sources said three soldiers were also killed in separate attacks in the eastern Libyan city yesterday.
Special forces there have been fighting militants from Ansar al Sharia, an extremist group the US blames for an attack on its Benghazi consulate in September 2012, in which the ambassador Christopher Stevens and three other Americans were killed.
Two years after a Nato-backed uprising ousted President Muammar Gaddafi, Libya's armed forces are struggling to contain militias, former fighters and Islamist militants who control parts of the oil-producing country.