Developing robust microwave-field sensors is both fundamentally and practically important with a wide range of applications from astronomy to communication engineering. The nitrogen vacancy (NV) center in diamond is an attractive candidate for such purpose because of its magnetometric sensitivity, stability, and compatibility with ambient conditions. However, the existing NV center-based magnetometers have limited sensitivity in the microwave band. Here, we present a continuous heterodyne detection scheme that can enhance the sensor's response to weak microwaves, even in the absence of spin controls. Experimentally, we achieve a sensitivity of 8.9 pT Hz-1/2 for microwaves of 2.9 GHz by simultaneously using an ensemble of nNV ~ 2.8 × 1013 NV centers within a sensor volume of 4 × 10-2 mm3. Besides, we also achieve 1/t scaling of frequency resolution up to measurement time t of 10,000 s. Our scheme removes control pulses and thus will greatly benefit practical applications of diamond-based microwave sensors.