Former Liverpool boss Benitez, who replaced Chelsea favourite Roberto Di Matteo in the Stamford Bridge dugout last week, was treated to a hostile reception from sections of the home crowd at his first game in charge, as his side drew against Manchester City on Sunday afternoon.
But Cech, who has recently been captaining the side in the absence of John Terry and Frank Lampard, insists that the players should remain fully in support of their new boss, regardless of any previous rivalry with Benitez.
"For all of [the players], it's the same; you get on with it or you're out," Cech told reporters. "So it's up to you how you react. It's as simple as that.
"You are here to play - to do as the manager and the club wants you to do. If you don't like it, then you can leave.
"Getting to know another new manager and his methods is not a problem. If you stay sad, you will end up on the bench even more disappointed."
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Cech also said that the chorus of boos from the home fans that greeted Benitez's arrival in the dugout had a minimal effect on his own performance, and insisted that an upturn in results is the most desirable outcome for all involved.
He added: "I wasn't listening to the crowd much. When I go on the pitch I am in my zone and concentrating on my own job.
"We know there is a history, but he has come here to improve the situation at the club and to help. He can only do that by getting good results and let's hope that is the case.
"It's up to us, as players, to help the fans move on. We need a good run of results for them to change their minds.
"I think the manager deserves to have a chance, as does every new manager who has joined the club. It's up to us to make things happen and to send the fans home happy.
"When you come to a situation where you change the manager, it's because the situation is bad and everybody is to blame.
"As players, we feel our share of responsibility because we are responsible for the manager leaving and we want to put things right for the new manager.
"We know the score - if the players are performing, nothing happens to the manager. But everybody knows our last few results were not good and the manager was changed.
"When you change the manager and his first game is against the champions, you want a positive start - and we have something to build on."