McMeikan, who will stay in his role until a successor is appointed, has headed up Greggs for four years.
Derek Netherton, chairman of the bakery chain, paid tribute to McMeikan this morning.
"We are very grateful to Ken for the valuable contribution he has made to Greggs. He has led the company through the major changes that have put us in a strong position for the future with a clear strategy for growth in a difficult environment," he said.
Brakes, which supplies food and machinery to the catering industry and has over 10,000 staff, is owned by private equity firm Bain Capital.