Some 1,000 Iraqi soldiers and a force of 2,500 marines, sailors and soldiers went house-to-house on the third day of their sweep of the restive far western Iraqi town of Husayba.
The sweep is coined Operation Steel Curtain and focuses on the upper Euphrates town, located in the Sunni Arab province of Al-Anbar.
The operation's goals "are to restore Iraqi sovereign control along the Iraqtroy the al Qaeda in Iraq terrorists operating throughout al Qaim region," the military said.
The 36 suspected insurgents killed include 17 who were reported dead after air strikes on Sunday, the military said, adding there were no reports of civilian casualties.
The US marine was killed "by enemy small arms fire" on Sunday while he was "conducting clearing operations in Husayba," the military said.
Militants loyal to al Qaeda in Iraq however threatened in an Internet statement to sharply intensify their campaign of violence unless the offensive ended.
The al Qaeda statement also promised to destroy the homes of all Iraqi soldiers and government employees in response to recent comments by Iraq's Defence Minister that those who sheltered insurgents in their homes would be considered targets.
The group said it is giving "the apostate government and its (US) master 24 hours to end their campaign against the Sunni people. After that they will only see from us the worst and something that's going to make the earth tremble under their feet."
The authenticity of the statement could not be independently verified.
Iraqi soldiers on Monday also shot and killed "three terrorists dressed in women's clothing near the entrance to the safety zone established for displaced persons."
The disguised trio "brandished weapons as they neared the checkpoint the Iraqi soldiers were manning, but were unable to use them before being killed by the soldiers."
The soldiers "identified the terrorists as foreign fighters," the statement read.
Separately, a US soldier in north-central Iraq attached to the US army's Task Force Band of Brothers was killed "by an improvised explosive device explosion late (Sunday) while on patrol," another statement read.
The deaths bring to at least 2,049 the number of US military personnel who have died in Iraq since the March 2003 invasion, according to an AFP tally based on the Iraq Coalition Casualty Count, an independent group that follows US casualties in Iraq.
Amid growing opposition to the war in Italy, Iraqi President Jalal Talabani implored Rome to keep its 3,000 troops in his country.
"A premature withdrawal would be a disaster for the Iraqi people and a victory for terrorism," Talabani wrote in a letter published in the La Stampa newspaper. The Iraqi leader began a six-day visit to Italy on Monday.
"Neither you nor the free and democratic world can allow yourselves to abandon the cause of democracy and leave Iraq to the terrorists."
An Australian special forces soldier was reported killed in an accident, while at least 23 Iraqis were killed in violence across the country, including six police officers killed in a sundown car bomb attack in southern Baghdad.
The blast, which targeted a police patrol in the Dura district, also killed three civilians and injured 10, including a number of policemen, a security source said.
In a separate incident in the same neighbourhood, two police were killed and two others wounded by an explosion while they were on patrol.
Two soldiers were killed and 14 injured in the late morning when a suicide bomber blew up his car at a checkpoint west of the northern oil city of Kirkuk, police said.
At sundown a police officer was shot dead in Kirkuk, local officials said.
Despite the wave of violence Iran hopes to resume sending pilgrims to holy sites in Iraq starting Tuesday under an agreement signed by the two governments in April - as long as the Iraqi embassy in Tehran issues the visas.
Under April's agreement 1,500 Iranian pilgrims per day were allowed travel to Iraq.
Hundreds of thousands of Iranian pilgrims have visited Iraq outside of any legal framework since the fall of Saddam Hussein's regime in 2003.
Several hundred have been arrested amid US suspicions that agents and militiamen are infiltrating as pilgrims in disguise.