"There were about four different incidents going on in Baghdad today, resulting in a total of three Iraqi police killed, six civilians killed, eight police wounded and nine civilians wounded," he said.
He also said a "total of 19 enemy" were killed in clashes involving US forces and followers of Shia leader Muqtada al-Sadr.
Eighteen of those were killed at the leader's stronghold in the Shia district of al-Sadr city when US forces moved to break up checkpoints put up by his forces there.
And in central Baghdad, four men wearing the black of al-Sadr's Mahdi Army fired at police from the top of a building.
Seizing key players
The clashes followed an operation hours earlier when US forces seized two key players in al-Sadr's Mahdi army, including a financier and a man responsible for eastern Baghdad operations, according to Kimmitt.
Fighting has gone on for days between al-Sadr's men and the US-led occupation in Baghdad and in southern Iraqi cities.
Al-Sadr is wanted over the murder of a rival leader last year.