At least 15 civilians were killed and 17 wounded when a suicide car bomb exploded outside the main police headquarters in the town of Baquba, north-east of Baghdad.
Police said the bomber tried to ram his car into the police station but was blocked by a concrete barrier and detonated his explosives near civilians instead.
In the northern city of Mosul, 15 people were killed and four wounded when the other suicide bomber blew himself up among a crowd of police officers in a hospital compound.
A large crater was blown in the road and at least five cars were destroyed. Most of the victims, if not all, were thought to be police officers waiting to collect their salaries.
"A lion in the martyrs' brigades of al Qaeda Organisation for Holy War in Iraq attacked a gathering of apostates seeking to return to the apostate police force in Mosul near the hospital," al Qaeda's Iraqi unit said in a statement posted on the Web site.
"The martyr was wearing an explosives belt and blew himself up after he entered the crowd."
A separate mortar attack on the city hall building in Mosul killed one person and wounded three.
Army of Ansar al-Sunna said it shot dead an Iraqi translator working for US forces and posted a video of the killing on the Internet. The video showed the hostage appealing to other translators not to deal with US forces before he was blindfolded and shot in the head.
More than a week after Iraq's first multi-party election in 50 years, the final result remains unknown, although partial results showed a coalition of Iraq's main two Kurdish parties has moved into second place in counting so far.
The leading Shia alliance has around 2.3 million votes, the Kurds have 1.1 million and Allawi's bloc has around 620,000.
Officials stressed the results did not necessarily give a clear picture of the final distribution of the votes.
Meanwhile, one of the key figures in the Shia alliance rejected calls for US-led troops to leave Iraq immediately.
"I think it is premature to ask the multinational forces to leave now," said Ibrahim Jaafari, head of the Dawa Party and a leading contender to be Iraq's next prime minister.
"If the multinational forces left now, Iraq could face a bloodbath. I believe this 100 percent," he told Reuters.
Speaking at a counter-terrorism conference in Saudi Arabia, Iraq's interim Interior Minister Falah al-Naqib said he believed the country could have full control of its internal security within 18 months.
Moreover, there was still no word on the fate of four Egyptian engineers who were kidnapped in Baghdad on Sunday as they left their home to work on an Iraqi mobile phone project.
An insurgent group that claims to have kidnapped the Italian journalist, Giuliana Sgrena, said on Monday it would "soon" decide her fate.
The group made no mention of a deadline it had set in an earlier statement that threatened to kill Sgrena by Monday if Italian troops did not leave Iraq.