Born in Yoichi, Hokkaidō, Japan, Mohri earned degrees in chemistry from Hokkaido University and a Doctorate from Flinders University in Adelaide, South Australia, in 1976.
Most of Mohri's work has been in the field of materials and vacuum sciences. From 1975 to 1985, Mohri was a member of the nuclear engineering faculty of Hokkaido University, where he worked on nuclear fusion-related projects.
Mohri was selected by the National Space Development Agency of Japan to train as a payload specialist for a Japanese materials science payload. He flew his first space mission aboard STS-47 in 1992 as chief payload specialist for Spacelab-J. Mohri subsequently made another trip into space as part of mission STS-99 in 2000.
As of 2007, Mohri is the Executive Director for the Miraikan (National Museum of Emerging Science and Innovation) in Tokyo.
On 16 March 2006 Mohri was appointed an Honorary Member of the Order of Australia (AM), "for service to Australia-Japan education and science relations".