"What we want to say to that person and, in fact, everyone else out there, is that as you sit down to enjoy Christmas dinner with your family to remember that Mr Voudouris had a family of his own, a family who would dearly love some closure to this tragedy.''
Mr Voudouris was found dead at the back entrance to Mykonos Pizza, the shop he owned and where he lived.
Police believe he was killed between 1.30am and 4.30am on June 18.
Mr Whitehead said "as time goes on, people's loyalties change'' and he urged anyone with information to come forward.
"The focus of the 20-strong investigation team remains the same as when we began this case six months ago, to identify those responsible and bring them before the courts,'' he said.
Anyone with information should call Paeroa Police on 07 862 8744 or Crime Stoppers on 0800 555 111.