This paper considers a single-hop wireless network where a central node (or fusion center, FC) collects data from a set of m energy harvesting (EH) nodes (e.g. nodes of a wireless sensor network). In each time slot, k of m nodes can be scheduled by the FC for transmission over k orthogonal channels. FC has no knowledge about EH processes and current battery states of nodes; however, it knows outcomes of previous transmission attempts. The objective is to find a low complexity scheduling policy that maximizes total throughput of the data backlogged system using the harvested energy, for all types (uniform, non-uniform, independent, correlated (i.e. Markovian), etc.) EH processes. Energy is assumed to be stored losslessly in the nodes batteries, up to a storage capacity (the infinite capacity case is also considered.) The problem is treated in finite and infinite problem horizons. A low-complexity policy, UROP (Uniformizing Random Ordered Policy) is proposed, whose near optimality is shown. Numerical examples indicate that under a reasonable-sized battery capacity, UROP uses the arriving energy with almost perfect efficiency. As the problem is a restless multi-armed bandit (RMAB) problem with an average reward criterion, UROP may have a wider application area than communication networks.