Locals had complained about the activities of the mayor of Yayladere district, which includes Yedisu, for "working in cooperation with the Turkish security forces, exercising threats and pressure harmful to the people", the PKK document said.
Akyurek, a Kurd and member of the ruling Justice and Development Party (AKP), was abducted on Wednesday during a visit to a mountainous part of his constituency.
The PKK denied the mayor had been released, as his son Zulfu Akyurek announced on Saturday after receiving an anonymous telephone call. Local AKP chairman Yusuf Coskun also told Turkish television that the mayor was freed in a rural zone hundreds of kilometres from where he was taken prisoner.
"The person in question will be freed in the case where he is found innocent by an inquiry that will be carried out into the allegations against him," the PKK statement said, adding the hostage was being held in a zone under PKK control and was in good health.
The PKK has increased attacks on the Turkish army in the country's southeast since ending a five-year unilateral ceasefire in June 2004 on grounds that Ankara's reforms to expand Kurdish freedoms were inadequate.
Earlier this month, the group abducted a Turkish soldier and was also blamed for a bomb attack that killed five people in a popular seaside resort.
The Kurdish conflict in Turkey has claimed about 37,000 lives since 1984, when the PKK took up arms against Ankara to fight for Kurdish self-rule.