The second bomb, hidden in a cooking pot, exploded near a government building in Thessaloniki, causing some damage but no injuries.
Phone warnings to Greek media were made before both blasts, in what appears to be a series of attacks against financial insitutions and government buildings.
Violence in Greece has escalated since police shot dead an Athens teenager last December.
The shooting sparked Greece's worst riots in decades.
While bomb attacks against foreign and domestic banks and other targets have become more frequent in recent months, they are usually organised to avoid causing injury.