Marine and hydrokinetic (MHK) energy from free-flowing waves, tides and currents comprises an important source of clean and renewable energy. The development of technologies for harnessing MHK energy and assessing its environmental effects is still at early stage. In this work, we develop a computational model for simulating hydrokinetic turbines with bed-load sediment transport under clear-water scour conditions. Turbulent flow is simulated using large-eddy simulation (LES). The bed surface elevation is simulated by solving the non-equilibrium continuity equation, the so-called Exner-Polya equation. The turbine blades are parameterized as actuator lines. The developed model is applied to simulate the flow past an axial-flow hydrokinetic turbine mounted on an erodible bed in an open channel. A good agreement with measurements is obtained. The effects of turbine operating conditions and sediment particle sizes on the bed-load sediment transport are then examined. Finally the turbine wake characteristics are investigated for a rigid flat bed and the eroded bed under different turbine operating conditions.