We consider the problem of downlink scheduling in an LTE network powered by energy harvesting devices. We formulate optimization problems that seek to optimize popular energy efficiency metrics subject to mandatory LTE network constraints along with energy harvesting causality constraints. We identify a key sub-problem pertaining to maximizing the weighted sum rate that is common for all optimization problems that we consider, and is also of independent interest. We show that the latter sub-problem can be reformulated as a constrained submodular set function maximization problem which enables us to design a constant-factor approximation algorithm for maximizing the weighted sum rate. We leverage this algorithm to design approximation algorithms (with provable performance guarantees) for maximizing all considered energy efficiency metrics over an energy harvesting LTE downlink. Our proposed algorithms are simple to implement and offer superior performance.