McKenzie had completed two years with Bangladesh after joining the Tigers during the tour of the West Indies in July 2018 but he was expected to continue in the role after agreeing an extension.
“We had a commitment with him for white-ball cricket,” BCB chief executive Nizamuddin Chowdhury told AFP.
“Before the COVID-19 outbreak, he agreed to work with us in all formats. But in the changed circumstances, with new COVID-19 protocols, he informed us recently that it was getting really difficult for him.
“He would have to come 14 days earlier now and had to spend even more time with Bangladesh away from his family, which was no longer possible for him.
“So he wanted to work in either red-ball or white-ball cricket. But we felt that it was better to discontinue the arrangement with him.”
McKenzie said it was a tough decision.
“It is difficult to leave a role which I have thoroughly enjoyed. From the very beginning I have been made to feel a part of the team and I am going with happy memories,” he said.
Bangladesh are expected to return to international cricket in October with a three-Test series against Sri Lanka after a break of more than five months, triggered by the coronavirus pandemic.
Bangladesh cricketers resumed training in July, but only one player and one trainer are allowed inside Dhaka’s main stadium at a time because of coronavirus restrictions.