The lawsuits allege Jagath Jayasuriya oversaw military units that attacked hospitals and killed, disappeared and tortured thousands of people in the final phase of Sri Lanka's civil war in 2009.
Jayasuriya has diplomatic immunity in Brazil and five other countries where he is ambassador - Colombia, Peru, Chile, Argentina and Suriname.
But the groups pursuing the suits hope they will compel regional governments to open investigations of Jayasuriya, remove his immunity and expel him.
Carlos Castresana Fernandez, the lawyer coordinating the effort, told the Associated Press news agency that suits have been filed on Monday in Brazil and Colombia.
Petitions will be filed in Argentina, Chile and Peru in the coming days, he said, adding that authorities in Suriname had refused to accept the petition.
"This is one genocide that has been forgotten, but this will force democratic countries to do something," Fernandez said. "This is just the beginning of the fight."
The criminal suits were spearheaded by the human rights group International Truth and Justice Project (ITJP), an evidence-gathering organisation based in South Africa.
The suits say Jayasuriya was commander of the Vanni Security Force from 2007 to 2009, one of the bloodiest periods in a 26-year war estimated to have killed more than 100,000 people.
Sri Lanka to get first Tamil military leader in 50 years
The UN estimates between 40,000 and 70,000 died in the final phase alone, when Sri Lanka's military defeated the separatist armed group, the Liberation Tigers of Tamil Eelam.
According to the suits, Jayasuriya oversaw an offensive from Joseph Camp in the northern town of Vavuniya.
ITJP said it interviewed 14 people who survived torture or sexual violence at the camp.
"There is no way General Jagath Jayasuriya can claim not to have known that torture routinely occurred in his camp; there were purpose built underground torture chambers, equipped with manacles, chains and pulleys for hoisting victims upside down," Yasmin Sooka, the ITJP's executive director, said in March.
"If the detainees could hear each other screaming at night from adjacent buildings, so could he."
Human rights groups have long been after Jayasuriya, but the Sri Lankan government has refused to try him or others allegedly involved in war abuses.
A few years after the war ended, he retired from the military. Jayasuriya was appointed ambassador to Brazil in 2015 and the other countries were added to his purview over the following two years.
The Sri Lankan army has denied committing war crimes.
Fernandez, the lawyer coordinating the lawsuits against Jayasuriya, has worked on international cases against war criminals in Guatemala, Argentina and Chile.
In the case of Chilean General Augusto Pinochet, he ended up being arrested and held for a time in England because of international lawsuits filed against him.