In this paper, the design of a wireless communication device relying exclusively on energy harvesting is considered. Due to the inability of rechargeable energy sources to charge and discharge at the same time, a constraint we term the energy half-duplex constraint, two rechargeable energy storage devices (ESDs) are assumed so that at any given time, there is always one ESD being recharged. The energy harvesting rate is assumed to be a random variable that is constant over the time interval of interest. A save-then-transmit (ST) protocol is introduced, in which a fraction of time ρ (dubbed the save-ratio) is devoted exclusively to energy harvesting, with the remaining fraction 1 - ρ used for data transmission. The ratio of the energy obtainable from an ESD to the energy harvested is termed the energy storage efficiency, η. We address the practical case of the secondary ESD being a battery with η < 1, and the main ESD being a super-capacitor with η = 1. The optimal save-ratio that minimizes outage probability is derived, from which some useful design guidelines are drawn. In addition, we compare the outage performance of random power supply to that of constant power supply over the Rayleigh fading channel. The diversity order with random power is shown to be the same as that of constant power, but the performance gap can be large. Furthermore, we extend the proposed ST protocol to wireless networks with multiple transmitters. It is shown that the system-level outage performance is critically dependent on the relationship between the number of transmitters and the optimal save-ratio for single-channel outage minimization. Numerical results are provided to validate our proposed study.