It came after the shock resignation of Dom McKay as the club’s chief executive on Friday after just 72 days in the job.
"We are all transient figures at this great club,” said Postecoglou. “The constant is the fans and they expect us to get on with it.
“I've done that throughout my career and will continue to. Everyone is allowed to express opinions in whatever way they choose to. Our job is to send the 60,000 here home happy and the millions around the world.
“Sometimes it's good to be kept on your toes and I am sure our supporters will do that.”
Postecoglou played down the influence McKay had on his appointment as manager in the summer and expressed his happiness at the prospect of now working under interim CEO Michael Nicholson.
"We have to be careful with the narrative of who brought me to the club,” added the Australian coach who took charge of Celtic in the summer after the club’s failed pursuit of former Bournemouth manager Eddie Howe.
“I was as surprised as anyone that Dom resigned but I am going to be respectful of him and his family. When people cite personal reasons, there's reasons for that. Michael is now in charge and that's good because I already know him.”