Wireless sensors can integrate rechargeable batteries and energy-harvesting (EH) devices to enable long-term, autonomous operation, thus requiring intelligent energy management to limit the adverse impact of energy outages. This work considers a network of EH wireless sensors, which report packets with a random utility value to a fusion center (FC) over a shared wireless channel. Decentralized access schemes are designed, where each node performs a local decision to transmit/discard a packet, based on an estimate of the packet's utility, its own energy level, and the scenario state of the EH process, with the objective to maximize the average long-term aggregate utility of the packets received at the FC. Due to the non-convex structure of the problem, an approximate optimization is developed by resorting to a mathematical artifice based on a game theoretic formulation of the multiaccess scheme, where the nodes do not behave strategically, but rather attempt to maximize a \emph{common} network utility with respect to their own policy. The symmetric Nash equilibrium (SNE) is characterized, where all nodes employ the same policy; its uniqueness is proved, and it is shown to be a local maximum of the original problem. An algorithm to compute the SNE is presented, and a heuristic scheme is proposed, which is optimal for large battery capacity. It is shown numerically that the SNE typically achieves near-optimal performance, within 3% of the optimal policy, at a fraction of the complexity, and two operational regimes of EH-networks are identified and analyzed: an energy-limited scenario, where energy is scarce and the channel is under-utilized, and a network-limited scenario, where energy is abundant and the shared wireless channel represents the bottleneck of the system.