A numerical method is developed for simulating unsteady, 3-D, laminar flow through a bileaflet mechanical heart valve with the leaflets fixed. The method employs a dual-time-stepping artificial-compressibility approach together with overset (Chimera) grids and is second-order accurate in space and time. Calculations are carried out for the full 3-D valve geometry under steady inflow conditions on meshes with a total number of nodes ranging from 4×105 to 1.6×106. The computed results show that downstream of the leaflets the flow is dominated by two pairs of counter-rotating vortices, which originate on either side of the central orifice in the aortic sinus and rotate such that the common flow of each pair is directed away from the aortic wall. These vortices intensify with Reynolds number, and at a Reynolds number of approximately 1200 their complex interaction leads to the onset of unsteady flow and the break of symmetry with respect to both geometric planes of symmetry. Our results show the highly 3-D structure of the flow; question the validity of computationally expedient assumptions of flow symmetry; and demonstrate the need for highly resolved, fully 3-D simulations if computational fluid dynamics is to accurately predict the flow in prosthetic mechanical heart valves.