Gunmen seized 18 employees of major Turkish construction firm Nurol Insaat on September 2 in the Sadr City area of northern Baghdad, where they were working on a football stadium project.
In a video posted online, men armed with submachine guns and wearing black uniforms, sunglasses and balaclavas stood behind men said to be the kidnapped Turks.
The men, who were dressed in grey and blue T-shirts, did not appear to have been harmed.
One of their demands was that Turkey order rebel forces to stop besieging four Shiite villages in northern Syria.
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This all indicates the militants are Shiite, but could also potentially be an attempt to mislead, and the group's makeup and provenance were not immediately clear.
The demands, addressed to Turkish President Recep Tayyip Erdogan, included Ankara stopping "the flow of militants from Turkey to Iraq," and "the passage of stolen oil from Kurdistan through Turkish territory."
"If Erdogan and his party do not respond, we will crush Turkish interests and their agents in Iraq by the most violent means," the group said.
Dozens of Turks have been kidnapped but later released in Iraq over the past 18 months by the Islamic State jihadist group, which overran large parts of the country last year.