Poyet, 39, was asked to stay away from Saturday's win over Millwall and boss Dennis Wise spoke to the team about the Uruguayan leaving.
"We wish him well. He texted everyone and said good luck," said acting assistant Alan Thompson. "He's a big part of what's happening here and he'll be missed."
Thompson added: "Leeds United is a big club and you have to get on with it.
"Gus has got great experience, he's a great coach, so he will be missed, but I think he is replaceable and things move on."