Nuri al-Maliki, the prime minister, said on Tuesday that "every militia which is loyal to a party is a militia".
"We must have one decision: When we say militia we mean all those who are armed other than the army and police."
Pressed to confirm that even the biggest militias run by governing parties would have to go, he specifically named the Kurdish peshmerga, the al-Mahdi Army of Shia cleric Muqtada al-Sadr and the Shia Badr movement as being among those that would have to be disbanded.
Al-Maliki also said he would overrule squabbling parties in his coalition and present parliament with his personal nominees for two key security posts if they fail to agree this week.
He added that if no consensus was found by the next session of parliament, due on Sunday, he would exercise his constitutional right to put his own nominees to a vote.
"This week is decisive for me. The next parliamentary session is decisive for me. I will offer candidates to parliament, with or without an agreement."
The worst bombing on Tuesday hit a bustling outdoor market north of Baghdad as Iraqis were doing their evening shopping in the Shia area of Hussainiya. At least 25 people were killed and 65 were wounded, interior ministry spokesman Lieutenant -Colonel Falah al-Mohamedawi said.
That attack came hours after a car packed with explosives blew up at a car showroom in the largely Shia city of Hilla, south of Baghdad, killing at least 12 people and wounding 32, Captain Muthana Khalid said.
The third bomb - hidden in a plastic bag - detonated outside a bakery in Baghdad, killing at least nine people and injuring 10, al-Mohamedawi said.
That explosion occurred at 9.15pm (0515 GMT) in New Baghdad, a mixed neighbourhood in the eastern part of the capital.
Elsewhere in Baghdad on Tuesday, mortar rounds fired by remote control from a car hit the third floor of the heavily guarded interior ministry and a nearby park, killing two government employees and wounding three other people.
A roadside bomb also killed one police officer and wounded four others in the capital, and police found nine bodies of people who had been shot in separate locations.
A decapitated body was discovered floating in the river about 50km south of the capital.
Police Captain Laith Mohammed, meanwhile, said a pregnant woman and her cousin were killed in uncertain circumstances in Samarra, as they were driving to a maternity hospital. When asked if they knew about the incident, the US military had no immediate comment.
Also on Tuesday, the US military said a roadside bomb killed an American soldier southeast of Baghdad and small arms fire killed a US soldier on Monday in Mosul.
The bodies of two marines missing after a helicopter crash in western Iraq over the weekend were also recovered.
Separately, the US military freed 204 male detainees from Abu Ghraib and other detention centres in Iraq on Tuesday, after the Iraqi-led Combined Review and Release Board reviewed their files and recommended release.
To date, the board has reviewed the cases of more than 39,000 detainees, recommending more than 19,600 individuals for release, the military said.