Altogether 11 bombs were detonated by remote control. The deadliest attack took place in the city of Hilla, 100 km (60 miles) south of Baghdad, when two car bombs blew up in quick succession, killing at least five people, police said.
It was not immediately clear who was behind the attacks, which appeared to be coordinated, but Sunni Islamist and other insurgents, including al Qaeda, have been regaining ground this year.
More than 6000 people have been killed in violence in 2013, reversing a decline in sectarian bloodshed that climaxed in 2006-07.
In Kut, four car bombs exploded separately, one of them near a primary school and one near a restaurant, killing at least two people and wounding 31, police said.
Leaflets signed by al Qaeda's Iraqi affiliate have been distributed on the streets of the Baquba in recent days, telling residents not to send their children to school or they will be killed, residents and police said.
Last week, a suicide bomber drove a truck packed with explosives into the playground of a primary school in northern Iraq and blew himself up, killing 14 children and their headmaster.
"The surge of violence in Iraq spares no one and no place," said a statement from the United Nations following that attack.