Cranmer resigned his position last week and will step down at the end of the year, with Meyers set to take over his new role on October 1.
Meyers, who will be based in Chicago, will be responsible for increasing Research International's profile in the US where its name is not as strong as it is in other markets.
Eric Salama, chief executive of Kantar, parent company to Millward Brown and Research International, said: "Research International has changed for the better over the past 18 months and for that we are grateful to Mark.
"It is more client-centric, more entrepreneurial and delivers more consistently to its clients. Now we are going to accelerate the process of change."
Salama said Research International and Millward Brown would "continue to occupy different spaces in the marketplace".
Meyers said: "I am delighted to become a part of one of the world's great research brands and look forward to working with a superb team to the benefit of Research International clients."
In April, Kantar announced the appointment of Eileen Campbell as the incoming chief executive of Millward Brown. She will assume sole charge of the company on October 1.
Kantar is one of the world's largest research networks and operates in 160 countries.