BAGHDAD (Reuters) - A series of car bombs exploded across Baghdad at rush hour on Tuesday, killing at least 24 people in the deadliest attacks since last month's election, from which votes are still being counted.

The apparently coordinated blasts targeted mainly Shi'ite districts of the capital.

"I was trying to pick up a passenger when a fireball and a thunderous blast shook my taxi," said Kadhim Ali, a driver who narrowly escaped one of three explosions in the slums of Sadr City. "I saw people with their clothes stained with blood shouting for help."

Preliminary results from the April 30 parliamentary election are expected within the coming days. Prime Minister Nuri al-Maliki, a Shi'ite, is seeking a third term despite worsening violence that critics blame in no small part on his policies towards the country's Sunni minority.

There was no immediate claim of responsibility for the bombings, but Sunni Islamist insurgents have been regaining ground in Iraq over the past year and are known to target Shi'ites.

There were blasts in several other neighborhoods, and three soldiers were killed by a roadside bomb while patrolling in Doura district, security sources said.

The past year has seen violence return to levels unseen since the darkest days of the U.S. military "surge" under President George W. Bush, with hundreds of civilians killed every month.

Government forces are fighting Sunni militants across western Anbar province, northern Iraq and in the countryside surrounding Baghdad. Shi'ite militia, once kept in check by Maliki and the Americans, have resurfaced to join the battle.

Further north, a suicide bomber driving a car blew himself up at an Iraqi army checkpoint in Rashad, around 40 km (25 miles) southwest of the disputed city of Kirkuk, killing five soldiers, security sources said.

(Reporting by Kareem Raheem and Raheem Salman; Editing by Mark Trevelyan)