We study transport in large, and strongly open, quantum dots, which might typically be viewed as lying well within the semiclassical regime. The low-temperature magnetoresistance of these structures exhibits regular fluctuations, with just a small number of dominant frequency components, indicative of the presence of dynamical tunneling into regular orbits. Support for these ideas is provided by the results of numerical simulations, which reveal wave function scarring by classically inaccessible orbits, which is found to persist even in the presence of a moderately disordered dot potential. Our results suggest that dynamical tunneling may play a more generic role in transport through mesoscopic structures than has thus far been appreciated.