A speaker for the captors, the Islamic Resistance Movement - Noman Brigades, said driver Medhat Chiwi would be killed unless his firm and a Kuwaiti company with which it works stop ferrying supplies to US forces in Iraq.
The group made their demand in a video aired by Al-Arabiya TV on Saturday.
"We ask the two companies to desist from these operations, [whose punishment is] beheading. We also warn all Turkish companies cooperating with occupation forces to pull out immediately," the speaker said.
An elderly bearded man was shown in the video, with a banner carrying the group's name in the background. Three hooded armed men were briefly shown.