The Assyrian Democratic Organization
The pre-dawn blasts hit churches across the capital over the course of an hour but there are no reports of casualties.
A mortar round also exploded at one of Baghdad’s hospitals, killing one.
Explosions have been heard in the rebel-held western city of Falluja where US planes and ground forces have been in action in recent days.
The Americans have, along with Iraqi forces, been trying to bring the restive Sunni city back under their control.
A delegation from the city offered to resume peace talks with the government earlier on Saturday if attacks were halted and the chief negotiator released. The current offensive began after they suspended talks earlier this week.
Christians targeted
No-one has said they were behind the church attacks which caused extensive damage.
The first bomb exploded at the church of Saint Joseph at about 0400 (0100 GMT) followed by similar explosions outside four other churches.
Flames engulfed the Roman Catholic church of St George in the central Baghdad district of Karrada.
“My family and I fled from the fire,” said caretaker Nabil Jamil.
Kamil Shabo, a 40-year-old labourer, said the bombers were “infidels”.
“They have no faith,” he told Reuters news agency. “It is a religious sanctuary, how could they attack a religious place?”
“Muslims and Christians have been living here in harmony for hundreds of years,” said Father Gabriel Shamami from St Joseph’s Church.
“I don’t think Iraqis would do this, especially during Ramadan.”
Monsignor Emmanuel Delly, the patriarch of the Chaldean Church, said there was nothing the tiny minority could do.
“If the government is powerless, what can we do,” he said. “We call on them [the attackers] not to touch the holy sites.”
The Association of Muslim Scholars in Iraq denounced the attacks against the churches, Arabic television station al-Jazeera reports.
There are around 750,000 Christians living in Iraq.
Many have fled to neighbouring countries since the rise of Islamic fundamentalism after the toppling of Saddam Hussein.
In August devices were set off at four churches in Baghdad and one in Mosul, killing a number of people and wounding dozens more.
