Plesiosaurians are an extinct group of highly derived Mesozoic marine reptiles with a global distribution that spans 135 million years from the Early Jurassic to the Late Cretaceous. During their long evolutionary history they maintained a unique body plan with two pairs of large wing-like flippers, but their locomotion has been a topic of debate for almost 200 years. Key areas of controversy have concerned the most efficient biologically possible limb stroke, e.g. whether it consisted of rowing, underwater flight, or modified underwater flight, and how the four limbs moved in relation to each other: did they move in or out of phase? Previous studies have investigated plesiosaur swimming using a variety of methods, including skeletal analysis, human swimmers, and robotics. We adopt a novel approach using a digital, three-dimensional, articulated, free-swimming plesiosaur in a simulated fluid. We generated a large number of simulations under various joint degrees of freedom to investigate how the locomotory repertoire changes under different parameters. Within the biologically possible range of limb motion, the simulated plesiosaur swims primarily with its forelimbs using an unmodified underwater flight stroke, essentially the same as turtles and penguins. In contrast, the hindlimbs provide relatively weak thrust in all simulations. We conclude that plesiosaurs were forelimb-dominated swimmers that used their hind limbs mainly for maneuverability and stability. 