Rendezvous is a fundamental operation in cognitive radio networks (CRNs) for establishing a communication link on a commonly-available channel between cognitive users. The existing works on rendezvous implicitly assume that each cognitive user is equipped with one radio (i.e., one wireless transceiver). As the cost of wireless transceivers is dropping, this feature can be exploited to significantly improve the rendezvous performance at low cost. In this study, we investigate the rendezvous problem in CRNs where cognitive users are equipped with multiple radios and different users may have different number of radios. We first study how the existing rendezvous algorithms can be generalized to use multiple radios for faster rendezvous. We then propose a new rendezvous algorithm, called role-based parallel sequence (RPS), which specifically exploits multiple radios for more efficient rendezvous. Our basic idea is to let the cognitive users stay in a specific channel in one dedicated radio and hop on the available channels with parallel sequences in the remaining general radios. We prove that our algorithm provides guaranteed rendezvous and derive the maximum time-to-rendezvous (TTR) and upper-bounds on the expected TTR. Extensive experiments are conducted to evaluate the proposed solutions.