We consider a Lévy-type jump process with time-dependent periodic coefficients. Assume that the process has a unique system of invariant probability measures. We obtain quantitative results, with explicit polynomial rates, for homogenization of the process. The result is new even in the time-independent framework, and in particular in this case it can be viewed as a quantitative version of our recent result for periodic homogenization of non-symmetric Lévy-type processes.