Hundreds of well-wishers and nearly as many journalists awaited the men as they arrived on a commercial flight from Hawaii in the port town of San Blas on Friday to a party atmosphere.
The men's 8-metre fibreglass boat ran into trouble off Mexico's Pacific Coast last November and drifted more than 8,000km before it was picked up by a Taiwanese trawler near the remote Marshall Islands two weeks ago.
Two other crew members, including the boat's captain, did not survive, due to their unwillingness to eat raw sea life, and were buried at sea, according to the men.
"I am really happy and I want to return to fishing," Lucio Rendon, one of the survivors, said.
The bewildered men were forced to field a barrage of questions from journalists ranging from whether they were drug smugglers to if they drank their own urine.
From poor backgrounds, the survivors have charmed Mexico with their tenacity on the voyage and reluctance to play the role of heroes on their return.
"Those who don't believe us, I hope they never have to go through what we went through"
Jesus Vidana, fisherman
But with no record of them leaving port some Mexican media suggested that they had been involved in a drug smuggling operation, picking up packages of cocaine at sea.
A government spokesman this week said the men were to be investigated for possible links to drug trafficking but Daniel Cabeza de Vaca, the attorney-general, said later there was no evidence so far that they were smugglers.
The men maintain they were fishing for sharks.
"Those who don't believe us, I hope they never have to go through what we went through," 27-year-old Jesus Vidana said.
Media have also said the three looked surprisingly healthy considering their ordeal, and suggested they may have survived by eating the flesh of their companions.
The men say they used a blanket as protection from the sun, and that their health improved dramatically on board the Taiwanese boat that rescued them near Kiribati on August 9.
The police commissioner on the Marshall Islands, George Lanwi, had voiced doubts on Wednesday about why the three men were not found earlier as they drifted along what he said was a busy trade route.
"How could they survive that distance in a boat with no canopy?" he asked. "It's suspicious. They look much healthier than we'd expect."
Jesus Vidana had to become
a fan of raw fish
On some days, the men lay still for hours to avoid drawing the attention of inquisitive sharks that circled and thumped against the boat.
At night, birds came to rest on the boat and would tuck their heads under their wings to sleep. Salvador Ordonez, the smallest of the three, became an expert at pouncing to catch them for food.
"My friend here is a cat," Vidana said of Ordonez.
While the three all say they want to return to fishing soon, they confirmed they have had offers for the rights to their story.