Graphene is an atomically thin material that features unique electrical and mechanical properties, which makes it an extremely promising material for future nanoelectromechanical systems (NEMS). Recently, basic NEMS accelerometer functionality has been demonstrated by utilizing piezoresistive graphene ribbons with suspended silicon proof masses. However, the proposed graphene ribbons have limitations regarding mechanical robustness, manufacturing yield, and the maximum measurement current that can be applied across the ribbons. Here, we report on suspended graphene membranes that are fully clamped at their circumference and have attached silicon proof masses. We demonstrate their utility as piezoresistive NEMS accelerometers, and they are found to be more robust, have longer life span and higher manufacturing yield, can withstand higher measurement currents, and are able to suspend larger silicon proof masses, as compared to the previous graphene ribbon devices. These findings are an important step toward bringing ultraminiaturized piezoresistive graphene NEMS closer toward deployment in emerging applications such as in wearable electronics, biomedical implants, and internet of things (IoT) devices.