The recording lasts about 17 minutes and includes urgent calls for an armored personnel carrier called a BearCat.
The officers who were shot worked for the Baton Rouge Police Department and the East Baton Rouge Sheriff's Office.
"There simply is no place for more violence," Gov. John Bel Edwards said. "That doesn't help anyone. It doesn't further the conversation. It doesn't address any injustice perceived or real. It is just an injustice in and of itself."
From his window, Joshua Godwin said he saw the suspect, who was wearing all black with a ski mask, combat boots and extra bullets. He appeared to be running "from an altercation."
Mike Spring awoke at a nearby house to a sound that he thought was from firecrackers. The noise went on for five to 10 minutes, getting louder.
Of the two officers who survived the shooting, one was hospitalized in critical condition, and the other was in fair condition. Another officer was being treated for non-life-threatening injuries, hospital officials said.
Officer Montrell Jackson posted his message on Facebook on July 8, just three days after the death of 37-year-old Alton Sterling, a black man killed by white Baton Rouge officers after a scuffle at a convenience store.
In the message, Jackson said he was physically and emotionally tired and complained that while in uniform, he gets nasty looks.
A friend of Jackson's family, Erika Green, confirmed the posting, which is no longer on Facebook. A screenshot of the image was circulating widely on the internet.
The Baton Rouge attack unfolded hours after a domestic violence suspect opened fire early Sunday on a Milwaukee police officer who was sitting in his squad car. The officer was seriously wounded, and the suspect fled and apparently killed himself, authorities said.
Police-community relations in Baton Rouge have been especially tense since Sterling's death. The killing was captured on cellphone video.
It was followed a day later by the shooting death of another black man in Minnesota, whose girlfriend livestreamed the aftermath of his death on Facebook. The next day, a black gunman in Dallas opened fire on police at a protest about the police shootings, killing five officers and heightening tensions even further.
Thousands of people have protested Sterling's death, and Baton Rouge police arrested more than 200 demonstrators.
Sterling's nephew condemned the killing of the three Baton Rouge officers. Terrance Carter spoke Sunday to The Associated Press by telephone, saying the family just wants peace.
"My uncle wouldn't want this," Carter said. "He wasn't this type of man.
A few yards from a police roadblock on Airline Highway, Keimani Gardner was in the parking lot of a warehouse store that would ordinarily be bustling on a Sunday afternoon. He and his girlfriend both work there. But the store was closed because of the shooting.
"It's crazy. ... I understand some people feel like enough is enough with, you know, the black community being shot," said Gardner, an African-American. "But honestly, you can't solve violence with violence."
Michelle Rogers and her husband drove near the shooting scene, but were blocked at an intersection closed by police.
"I can't explain what brought us here," she said. "We just said a prayer in the car for the families."