In 1997, Kontsevich gave a canonical formula for the quantization of arbitrary Poisson brackets. The formula involves an infinite collection of universal constants, given by high-dimensional integrals that are notoriously difficult to compute. I will describe forthcoming joint work with Peter Banks and Erik Panzer, in which we give an algorithm for the evaluation of these integrals as rational linear combinations of special transcendental numbers, called multiple zeta values (generalizations of special values of the Riemann zeta function). The approach is based on Francis Brown's work on periods of mixed Tate motives, particularly the moduli space of genus zero curves.