Previous works on cognitive relay networks (CRNs) considered only symmetric fading channels. However, in practical wireless propagation scenarios, it is likely that the channels of the secondary user (SU) and primary user (PU) may undergo different fading characteristics. In this paper, we assume that the channels of the secondary network (SU-source→SU-relay→SU-destination) are subject to Rician fading, whereas the channels of the link from the SU to the PU experience Rayleigh fading. Based on this framework, the end-to-end outage probability (OP) of CRNs is investigated for two different relaying schemes: i) in the absence of the direct link with decode-and-forward (DF) protocol and ii) in the presence of the direct link with incremental DF protocol. In particular, we derive both exact and asymptotic OP expressions for the considered CRNs. Our analysis reveals important insights into the impact of fading parameters on the CRN performance under distinct fading distributions.