The number of the wounded in the pair of attacks rose to 94, said the same source.
A suicide bomber detonated his explosive vest at a popular restaurant in New Baghdad district, southeastern Baghdad, killing 30 people and wounding some 66 others, the source told Xinhua on condition of anonymity.
Another suicide bomber blew up an explosive vest at the crowded market of al-Arabi in downtown Baghdad, leaving six people killed and 28 others wounded, the source added.
No group has so far claimed responsibility for the attacks but the Islamic State in Iraq and the Levant (ISIL) militant group, in most cases, was responsible for such suicide attacks in the country.
The attacks came on the day when the Iraqi government will lift the years-long night-time curfew imposed on Baghdad, and disarm some of Baghdad's districts of Kadhmiyah, Adhamiyah, Mansour, Karrada and Saidiyah.
The government's decision to ease security measures was seen as a sign of security improvement in Baghdad, after about six months of the appointment of Haider al-Abadi as prime minister.
Iraq has been gripped by some of the worst violence and terrorist attacks in years, killing at least 12,282 civilians and injuring 23,126 others in 2014.