We present a new theory for the gravitational-wave signatures of core-collapse supernovae. Previous studies identified axisymmetric rotating core collapse, core bounce, postbounce convection, and anisotropic neutrino emission as the primary processes and phases for the radiation of gravitational waves. Our results, which are based on axisymmetric Newtonian supernova simulations, indicate that the dominant emission process of gravitational waves in core-collapse supernovae may be the oscillations of the protoneutron star core. The oscillations are predominantly of mode character, are excited hundreds of milliseconds after bounce, and typically last for several hundred milliseconds. Our results suggest that even nonrotating core-collapse supernovae should be visible to current LIGO-class detectors throughout the Galaxy, and depending on progenitor structure, possibly out to megaparsec distances.