The Imam Ali Brigade gave the Turkish government a week to meet its demands, which included "banning Turkish companies from ferrying material to US bases in Iraq".
Video footage from the group aired by Aljazeera on Thursday showed the alleged hostage against a wall and holding up an identity document. He was named as Hasan Eskinutlu. The nationality of the translator was not specified.
Scores of Turks, mostly truck drivers but some entrepreneurs, have been targeted for kidnapping by armed Iraqi groups since the US-led invasion of Iraq in 2003. Most were subsequently released.
In another development, US and Iraqi forces detained the head of Karbala's local council after raiding his home early on Thursday, Iraqi officials said.
The US military initially said it could not confirm the reported arrest of Akeel al-Zubaidi, head of the provincial council in Karbala, 110km south of Baghdad.
It later issued a statement, saying Iraqi forces backed by US-led forces captured a "high-ranking terrorist network commander", naming him as Sheikh Aqeel but making no reference to his surname or any position on Karbala's provincial council.
"Aqeel commands a Kerbala terrorist network and is wanted for assassinating Iraqi citizens and planning and ordering attacks against Iraqi and coalition forces," the US military said.
A US military statement said Sheikh Aqeel provided financial support to sub-ordinate leaders and supplied them with improvised explosive devices (IEDs), or roadside bombs.
"The IEDs supplied by Aqeel are linked to the deaths of at least six coalition soldiers in 2005 and the deaths of a coalition soldier and an interpreter on June 8," the statement said.
Akeel al-Khazali, governor of the province of Karbala, told Reuters the raid took place at about 1am (2100 GMT) and said US and Iraqi forces had carried out the operation.
Zubaidi belongs to the Shia Fadhila party, a member of Nuri al-Maliki's ruling Shia Alliance coalition. Khazali said a security plan was in place in Karbala to prevent possible violence following the raid.
A spokesman for Fadhila said the arrest was "illegal" and demanded an investigation.
"The house of the Sheikh Akeel al-Zubaidi was brutally raided by the US and Iraqi forces," Hassan al-Shemari told a news conference in Baghdad.