The first bombs exploded near the market in the town of Balad Ruz, 45 miles northeast of Baghdad, killing 11 people and wounding 50, police and hospital officials said.
A suicide car bomber later attacked a checkpoint manned by Shiite militiamen near the city of Samarra, killing eight Shite fighters and wounding 15, authorities said.
Samarra and surrounding areas have been under constant attacks by the Islamic State group, which holds about a third of Iraq and neighboring Syria in its self-declared caliphate.
A second suicide bomber attacked another Shiite militia checkpoint just south of Samarra, killing eight fighters and wounding 16, police and hospital officials said.
More From This Section
All officials spoke on condition of anonymity because they were not authorized to talk to journalists.
No one immediately claimed the attacks, though Iraq sees near-daily attacks often claimed by the Islamic State group. Meanwhile in Baghdad, Iraqi Prime Minister Haider al-Abadi vowed to track down and punish those who smashed rare ancient artifacts in the northern city of Mosul.
The destruction is part of a campaign by the extremists, who have destroyed a number of shrines since last summer. They are also believed to have sold ancient artifacts on the black market to finance their bloody campaign.
"Those barbaric, criminal terrorists are trying to destroy the heritage of the mankind and Iraq's civilization," al-Abadi said. "We will chase them in order to make them pay for every drop of blood shed in Iraq and for the destruction of Iraq's civilization."