We propose HyDICE, Hybrid Discrete Continuous Exploration, a multi-layered approach for hybrid-system falsification that combines motion planning with discrete search and discovers safety violations by computing witness trajectories to unsafe states. The discrete search uses discrete transitions and a state-space decomposition to guide the motion planner during the search for witness trajectories. Experiments on a nonlinear hybrid robotic system with over one million modes and experiments with an aircraft conflict-resolution protocol with high-dimensional continuous state spaces demonstrate the effectiveness of HyDICE. Comparisons to related work show computational speedups of up to two orders of magnitude.