On Saturday, at a 40-year reunion for the American team, Burleson remembered the chilling events that led to the killings and how close he was to the Israeli hostages before they died.
Burleson recalled seeing a long line when he tried to return to a building where the athletes were staying.
"There were like 300 to 400 athletes there and they were saying move toward the front and you have to have your three pieces of ID ready," Burleson said. "I said, 'This is going to take forever.' "
Burleson was with two Italian basketball players and the three decided to take a shortcut through a garage across the street to return to their rooms.
"I got stopped by what I guess were German police, and they had a uniform," Burleson said. "Another came up and flashed a gun, and I said I was going to my room."
At that point, Burleson was not sure what was going on.
"By the time I got to the garage door, another German soldier who spoke English as well as anybody here said, 'Son, you are in the wrong place at the wrong time. We are in the process of bringing the hostages out right now.' "
Burleson said he saw the two Italian players on the ground with guns to their backs and he had a rifle pointed at his back.
"I could hear them start to bring the hostages out," he said. "The guy in a green leader suit and black mask, he stepped around the corner, and I looked at him and we were probably 60 feet away."
Burleson said as he looked at him, the German soldier removed the gun from his back and placed it at the back of his head and ordered him to face the wall.
Burleson remembers looking right into the wall.
"And I still see the blemishes today in that wall," he said. "And I started praying to God to allow me to get out of this situation and get back to the room where I needed to be."
And then he heard things that will never fade from his mind.
"The terrorists started bringing in the nine Israeli hostages by me, and I could hear the shuffling of their feet and could hear them crying; I could hear them crying." he said. Two Israeli members had already been killed before the nine hostages were brought out.
At that point, while relaying the story during a news conference Saturday, Burleson began to cry before sharing one final thought.
"They were going to die. In 45 minutes they were going to die," he said. "I could hear these men, these athletes, as they walked by, they loaded them, they take them to the helicopter. They release me, I turned, standing, go to the Olympic village and see the helicopter fly off. From there you know what happened."
The hostages were all killed when gunfire erupted at the airport, where five Palestinian terrorists and a West German policeman also died.
After Burleson told his story, there was silence in the room where the news conference took place.
"They would die then," he said. "I can hear them in my sleep."
Burleson excused himself, sobbing as he recalled the moment.
At that point, the news conference ended.
Contact Marc Narducci at 856-779-3225 or firstname.lastname@example.org, or follow on Twitter @sjnard.