Many animals that swim or fly use their body to accelerate the fluid around them, transferring momentum from their flexible bodies and appendages to the surrounding fluid. The kinematics that emerge from this transfer result from the coupling between the fluid and the active and passive material properties of the flexible body or appendages. To elucidate the fundamental features of the elastohydrodynamics of flexible appendages, recent physical experiments have quantified the propulsive performance of flexible panels that are actuated on their leading edge. Here we present a complementary computational study of a three-dimensional flexible panel that is heaved sinusoidally at its leading edge in an incompressible, viscous fluid. These high-fidelity numerical simulations enable us to examine how propulsive performance depends on mechanical resonance, fluid forces, and the emergent panel deformations. Moreover, the computational model does not require the tethering of the panel. We therefore compare the thrust production of tethered panels to the forward swimming speed of the same panels that can move forward freely. Varying both the passive material properties and the heaving frequency of the panel, we find that local peaks in trailing edge amplitude and forward swimming speed coincide and that they are determined by a non-dimensional quantity, the effective flexibility, that arises naturally in the Euler–Bernoulli beam equation. Modal decompositions of panel deflections reveal that the amplitude of each mode is related to the effective flexibility. Panels of different material properties that are actuated so that their effective flexibilities are closely matched have modal contributions that evolve similarly over the phase of the heaving cycle, leading to similar vortex structures in their wakes and comparable thrust forces and swimming speeds. Moreover, local peaks in the swimming speed and trailing edge amplitude correspond to peaks in the contributions of the different modes. This computational study of freely swimming flexible panels gives further insight into the role of resonance in swimming performance that is important in the engineering and design of robotic propulsors. Moreover, we view this reduced model and its comparison to laboratory experiments as a building block and validation for a more comprehensive three-dimensional computational model of an undulatory swimmer that will couple neural activation, muscle mechanics and body elasticity with the surrounding viscous, incompressible fluid.