The first blast took place near a police patrol in the suburb of Radwaniya, a largely Sunni neighborhood southwest of Baghdad, killing two policemen and wounding five.
Police also reported a roadside bombing on a commercial street in the Hay Jami'a neighborhood, which killed three people and wounded nine.
Near the Baghdad International Airport, police say a car bomb tore through a parking lot at one of the checkpoints leading to the airport, wounding five people and causing extensive damage to cars.
The statement, which identified the attacker as a man named Abu Muawiya al-Falluji, could not immediately be verified but it was posted on a website frequently used by the group.
Hospital officials confirmed the casualties. All spoke on the condition of anonymity because they are not authorized to brief the media.