We report measurements of dynamical tunneling rates of a Bose-Einstein condensate across a barrier in classical phase space. The atoms are initially prepared in quantum states that extend over a classically regular island region. We focus on the specific system of quantum accelerator modes of the kicked rotor in the presence of gravity. Our experimental data is supported by numerical simulations taking into account imperfections mainly from spontaneous emission. Furthermore, we predict experimentally accessible parameter ranges over which direct tunneling could be readily observed if spontaneous emission was further suppressed. Altogether, we provide a proof-of-principle for the experimental accessibility of dynamical tunneling rates in periodically driven systems.