The mechanism of core-collapse supernova explosions must draw on the energy provided by gravitational collapse and transfer the necessary fraction to the kinetic and internal energy of the ejecta. Despite many decades of concerted theoretical effort, the detailed mechanism of core-collapse supernova explosions is still unknown, but indications are strong that multi-D processes lie at its heart. This opens up the possibility of probing the supernova mechanism with gravitational waves, carrying direct dynamical information from the supernova engine deep inside a dying massive star. I present a concise overview of the physics and primary multi-D dynamics in neutrino-driven, magnetorotational, and acoustically driven core-collapse supernova explosion scenarios. Discussing and contrasting estimates for the gravitational-wave emission characteristics of these mechanisms, I argue that their gravitational-wave signatures are clearly distinct and that the observation (or non-observation) of gravitational waves from a nearby core-collapse event could put strong constraints on the supernova mechanism.