Rendezvous is a fundamental operation for cognitive users to establish communication links so as to realize data communications and network management. Most of existing rendezvous algorithms implicitly assume that each cognitive user is equipped with one radio, i.e., one wireless transceiver. As the cost of wireless transceivers is dropping, it becomes economically feasible to utilize multiple radios to significantly improve the rendezvous performance. In this paper, we propose an Adjustable Multi-Radio Rendezvous (AMRR) algorithm which exploits multiple radios for fast rendezvous based on available channels only. Suppose that a cognitive user is equipped with m radios. Our basic idea is to partition the radios into two groups: k stay radios and (m-k) hopping radios. The user stays on specific channels in the stay radios while hops on its available channels parallelly in the hopping radios. We prove that the maximum time-to-rendezvous (MTTR) of AMRR is upper-bounded by O(|C-1||C-2|/m-1m-2}), where |C-1| and |C-2| are the numbers of available channels of two users and m-1 and m-2 are the numbers of radios of the two users. This bound meets the lower bound of MTTR of any deterministic rendezvous algorithm when two users are equipped with the same number of radios (i.e., m-1=m-2). AMRR is adjustable in giving its best performance on either MTTR or E(TTR) by adjusting value of k. Simulation results show that AMRR performs better than the state-of-the-art.