Electromagnetic energy harvesting, i.e., capturing energy from ambient microwave signals, may become an essential part in extending the battery lifetime of wearable devices. Here, we present a design of a microwave energy harvester based on a cut-wire metasurface with an integrated PN junction diode. The cut wire with a quasistatic electric-dipole moment is designed to have a resonance at 6.75 GHz, leading to a substantial cross-section for absorption. The external microwaves create a unidirectional current through the rectifying action of the integrated diode. Using an electrical-circuit model, we design the operating frequency and the resistive load of the cut wire. Subsequently, by optimizing our design using full-wave numerical simulations, we obtain an energy harvesting efficiency of 50% for incident power densities in agreement with the typical power density of WiFi signals. Finally, we study the effect of connecting adjacent unit cells of the metasurface in parallel by a thin highly inductive wire and we demonstrate that this allows for the collection of current from all individual cells, while the microwave resonance of the unit cell is not significantly altered, thus solving the wiring problem that arises in many nonlinear metamaterials.