A statement publicised on Monday from a group calling itself the Green Brigade said Italian people had five days to demonstrate or the three hostages would be killed.
The statement, received by Dubai-based al-Arabiya television, which aired footage of the trio, said the men were kidnapped because they were "spies".
Footage was shown in the video of the three, sitting behind a small table and eating from a large pot with their fingers. The hostages were bearded and their faces looked drawn. It was not clear when the video was made.
One of four Italian security guards kidnapped by the Iraqi group on 12 April was executed two days later, and the others were threatened with death if Italy failed to withdraw its troops from Iraq.
Italy maintains a 3000-strong battalion in Iraq serving under the US occupation command.