The attacks killed 26 people and wounded another 53. Five bank guards were among the victims, according to Hussein al-Uzri, the Trade Bank of Iraq's chairman,
The bank is the conduit for much of the government's foreign-exchange transactions and its dealings with investors.
It is one of the public sector's most active financial institutions and has been working to encourage foreign investment in Iraq.
But the Islamic State of Iraq claimed the Trade Bank of Iraq was involved in "the robbery of the country's oil revenues and overseas funds" that were then transferred "to the pockets of senior government officials" and Shia Muslim parties.
The group had earlier claimed responsibility for a highly co-ordinated assault on Iraq's central bank by armed men and suicide bombers on June 13, the eve of the first session of Iraq's new parliament.
Eighteen people died in that attack.
Banks have become frequent targets for both criminals and fighters in recent months.
Armed men stole $6.5m from a Baghdad bank last summer.