Wireless sensor network (WSN) underpinning the smart-grid Internet of Things (SG-IoT) has been a popular research topic in recent years due to its great potential for enabling a wide range of important applications. However, the energy consumption (EC) characteristic of sensor nodes is a key factor that affects the operational performance (e.g., lifetime of sensors) and the total cost of ownership of WSNs. In this paper, to find the modulation techniques suitable for WSNs, we investigate the EC characteristic of continuous phase modulation (CPM), which is an attractive modulation scheme candidate for WSNs because of its constant envelope property. We first develop an EC model for the sensor nodes of WSNs by considering the circuits and a typical communication protocol that relies on automatic repeat request (ARQ)-based retransmissions to ensure successful data delivery. Then, we use this model to analyze the EC characteristic of CPM under various configurations of modulation parameters. Furthermore, we compare the EC characteristic of CPM with that of other representative modulation schemes, such as offset quadrature phase-shift keying (OQPSK) and quadrature amplitude modulation (QAM), which are commonly used in communication protocols of WSNs. Our analysis and simulation results provide insights into the EC characteristics of multiple modulation schemes in the context of WSNs; thus, they are beneficial for designing energy-efficient SG-IoT in the beyond-5G (B5G) and the 6G era.