In nearly half of the heart valve replacement surgeries performed annually, surgeons prefer to implant bileaflet mechanical heart valves (BMHV) because of their durability and long life span. All current BMHV designs, however, are prone to thromboembolic complications and implant recipients need to be on a life-long anticoagulant medication regiment. Non-physiologic flow patterns and turbulence generated by the valve leaflets are believed to be the major culprit for the increased risk of thromboembolism in BMHV implant recipients. In this paper, we review recent advances in developing predictive fluid-structure interaction (FSI) algorithms that can simulate BMHV flows at physiologic conditions and at resolution sufficiently fine to start probing the links between hemodynamics and blood-cell damage. Numerical simulations have provided the first glimpse into the complex hemodynamic environment experienced by blood cells downstream of the valve leaflets and successfully resolved for the first time the experimentally observed explosive transition to a turbulent-like state at the start of the decelerating flow phase. The simulations have also resolved a number of subtle features of experimentally observed valve kinematics, such as the asymmetric opening and closing of the leaflets and the leaflet rebound during closing. The paper also discusses a future research agenda toward developing a powerful patient-specific computational framework for optimizing valve design and implantation in a virtual surgery environment.