5G and cellular networks would become 1.4% contributors to the carbon footprint, almost on par with 2% of the aviation industry, and is only on the trajectory of further increasing their carbon footprint. Wireless basestations are one of the major contributors to the operational carbon footprint, as a consequence of transmitting at high power levels to achieve the required communication range and throughput. In order to further keep up with the increasing data rates, and more users getting connected, typical approaches make the existing base stations even more sophisticated, like going to massive MIMO architectures, or mmWave communications, the earlier requiring many more antennas and electronics, and the latter operating at high mmWave frequencies where power amplifiers are difficult to design. Instead, much simpler hardware can be used at lesser transmit power requirements if the base station deployment is densified, with the key idea being that numerous simpler base stations can equivalently do the job of a single sophisticated base station. This dense deployment also ends up saving the power cost of transmitting at higher signal levels, since the smaller base stations need to blast lesser power over the air. The key reason is that this dense network of base stations suffers from less environmental attenuation due to shorter links, thus requiring significantly less power to offset these environmental losses. Further, this densification strategy can be a scalable way of achieving the next generation cellular capacity, where networks can be densified as response to user demand. However, doing so would require addressing numerous challenges and opportunities in designing such networks, articulating the architecture, algorithm, and deployment-level challenges to drive the next decade of research toward greener cellular networks.