Wireless energy harvesting is regarded as a promising energy supply alternative for energy-constrained wireless networks. In this paper, a new wireless energy harvesting protocol is proposed for an underlay cognitive relay network with multiple primary user (PU) transceivers. In this protocol, the secondary nodes can harvest energy from the primary network (PN) while sharing the licensed spectrum of the PN. In order to assess the impact of different system parameters on the proposed network, we first derive an exact expression for the outage probability for the secondary network (SN) subject to three important power constraints: 1) the maximum transmit power at the secondary source (SS) and at the secondary relay (SR), 2) the peak interference power permitted at each PU receiver, and 3) the interference power from each PU transmitter to the SR and to the secondary destination (SD). To obtain practical design insights into the impact of different parameters on successful data transmission of the SN, we derive throughput expressions for both the delay-sensitive and the delay-tolerant transmission modes. We also derive asymptotic closed-form expressions for the outage probability and the delay-sensitive throughput and an asymptotic analytical expression for the delay-tolerant throughput as the number of PU transceivers goes to infinity. The results show that the outage probability improves when PU transmitters are located near SS and sufficiently far from SR and SD. Our results also show that when the number of PU transmitters is large, the detrimental effect of interference from PU transmitters outweighs the benefits of energy harvested from the PU transmitters.