Cloud Radio Access Network (C-RAN) disaggregates the functionalities of the base station in a way that some of the radio processing tasks are centralized in a virtualized computer pool of general-purpose processors (GPPs) on a cloud platform. This enables efficient utilization of the computational resources based on the spatio-temporal traffic fluctuations at cell sites. In this paper, we attempt to further reduce the computation resources by C-RAN on the cloud platform. First, we profiled the energy consumed in an OpenAirInterface (OAI) based C-RAN system using the existing Linux CPU frequency scaling governors. Based on the observations, we propose a traffic-aware compute resource tuning (CRT) scheme that reduces the energy consumption of C-RANs. The CRT scheme opportunistically lowers Modulation Coding Scheme (MCS) used while serving users by utilizing all of the available radio resources in every scheduling interval during non-peak hours. This reduction in the MCS helps in reducing energy consumption (due to usage of lower CPU clock frequency in the GPPs of the cloud platform) and fronthaul bandwidth requirements. Another benefit of the CRT scheme is its ability to work with any MAC scheduler. The extensive simulation results show how the CRT outperforms the existing frequency scaling governors in energy consumption while reducing fronthaul bandwidth requirements. © 2021 IEEE.