A fully three-dimensional potential flow Boundary Element Method for the analysis of the unsteady flow around marine current turbines is presented. An empirical vortex model independent of the induced velocities is assumed for the turbine wake. The application of the method to the analysis of a controllable pitch horizontal axis marine current turbine is illustrated for straight and yawed inflow conditions at two different pitch settings in a wide range of tip-speed-ratios. Comparison of the numerical calculations with experimental data available in the literature is presented. The effect of a tidal velocity profile on the unsteady turbine blade is illustrated.