This paper considers the use of energy harvesters for cooperative relaying in a large relay network, which consists of N energy-harvesting (EH) relays and one source-destination pair. In particular, a threshold-based 'save-then-transmit' scheme is employed at the relays, where each relay transmits only when both the backward and forward link channel coefficients are above certain thresholds. We assume that the time scale of EH is much larger than that of communication blocks. For general channel fading models, we derive the asymptotic average throughput for the case with many relays, by using the amplify-and-forward (AF) relaying scheme. The throughput maximization is cast as a joint optimization problem over the transmission thresholds corresponding to all possible harvested energy rate states, which is shown to be non-convex in general. By applying a convexification technique via randomization, the original problem is transformed into a new formulation with a generalized threshold-based transmission scheme, which is shown to be efficiently solvable by bisection search, with the help of an offline look-up table only related to the channel statistics. Finally, with some numerical experiments, we demonstrate the performance gain of the proposed threshold-based transmission scheme against some suboptimal ones.