Devices and method are disclosed for reducing pressure in a patient's vena cava and in the right atrium of a patient's heart. A valve is implanted in the inferior vena cava (IFC) between the patient's renal veins and hepatic veins. When pressure in a patient's IFC reaches high levels, i.e., the pressure differential across the valve reaches high levels, the valve automatically closes or begins to close. The closing is caused by the pressure of the patient's blood moving one or more flaps in the valve to a closed or near-closed position. This prevents excessive pressure in the upper or superior vena cava and the right atrium of the patient's heart. The device also includes a pressure relief feature that opens and allows retrograde flow of blood if the pressure differential exceeds a yet higher pressure.