Leach will start a new role as head of health, housing and environmental policy research at the think tank after working for just over 18 months at Capacitybuilders.
A Capacitybuilders spokeswoman said the organisation had not yet decided whether a permanent new chief executive would be appointed.
Interim arrangements would be made before the end of the month, and, in the meantime, Stephen Dunmore, the organisation’s chair, would take a more active role, she said.
Capacitybuilders’ funding from the Office for Civil Society will run out in March and the spokeswoman said she did not yet know whether it would be renewed.
Leach was unavailable for interview but, in a statement, he said: "The role at ResPublica was too good to turn down."
Through a spokeswoman, he denied that uncertainties over government funding for Capacitybuilders had influenced his decision to leave.
Dunmore said: "I am obviously sorry to see Matt go, particularly at a time of so much change and uncertainty for the sector."