About 50,000 people, including former Sunni fighters, have reportedly signed up to the units.
They are paid about $10 a day by the US military to man checkpoints in their own communities.
On Thursday, the leaders of four Awakening units were killed by in a drive-by shooting in the town of Rabia in northwest Iraq near the Syrian border, police said.
Crowded area
Witnesses in Muqdadiya said a woman walked up to the building, in a street full of shops, and began asking questions, the Reuters news agency reported.
She detonated the vest she was wearing when people who were out shopping before Friday prayers began gathering around her.
"We saw several bodies. It is Friday and the area was crowded," Ammar Fadhel said.
A similar attack on November 27 by a female suicide bomber wounded seven US soldiers and five Iraqi civilians in Baquba which, like Muqdadiya, is in Diyala province.