This study employs extensive three-dimensional direct numerical simulations to investigate the hemodynamics around a bileaflet mechanical heart valve. In particular, this study focuses on assessing whether non-Newtonian rheological behaviors of blood, such as shear-thinning and yield stress behaviors, exert an influence on hemodynamics compared to the simplistic Newtonian behavior under both steady inflow and physiologically realistic pulsatile flow conditions. Under steady inflow conditions, the study reveals that blood rheology impacts velocity and pressure field variations, as well as the values of clinically important surface and time-averaged parameters like wall shear stress (WSS) and pressure recovery. Notably, this influence is most pronounced at low Reynolds numbers, gradually diminishing as the Reynolds number increases. For instance, surface-averaged WSS values obtained with the non-Newtonian shear-thinning power-law model exceed those obtained with the Newtonian model. At Re=750, this difference reaches around 67%, reducing to less than 1% at Re=5000. Correspondingly, pressure recovery downstream of the valve leaflets is lower for the shear-thinning blood than the constant viscosity one, with the difference decreasing as the Reynolds number increases. On the other hand, in pulsatile flow conditions, jets formed between the leaflets and the valve housing wall are shorter than steady inflow conditions. Additionally, surface-averaged wall shear stress and blood damage (BD) parameter values are higher (with differences more than 13% and 47%, respectively) during the peak stage of the cardiac cycle, especially for blood exhibiting non-Newtonian yield stress characteristics compared to the shear-thinning or constant viscosity characteristics. Therefore, blood non-Newtonian behaviors, including shear-thinning and yield stress behaviors, exert a considerable influence on the hemodynamics around a mechanical heart valve. All in all, the findings of this study demonstrate the importance of considering non-Newtonian blood behaviors when designing blood-contacting medical devices, such as mechanical heart valves, to enhance functionality and performance.