The threat was contained in a video statement issued by the previously unknown Brigades of the Swords of the Right.
The New Zealand family of Harmeet Singh Sooden, a Canadian citizen, were waiting in Auckland for news.
The other three hostages were Canadian James Loney, American Tom Fox and Briton, Norman Kember.
Meanwhile, 11 Iraqi troops were killed in a bomb attack and ensuing gunfire in Adhaim, 100 kilometres north of Baghdad.