Nodes in a cognitive radio mesh network comprised of secondary users may select from a set of available channels provided they do not interfere with primary users. This ability can improve overall network performance but introduces the question of how best to use these channels. Given a routing multipath M, we would like to choose which channels each link in M should use and a corresponding transmission schedule so as to maximize the end-to-end data flow rate (throughput) supported by the entire multipath. This problem is relevant to applications such as streaming video or data where a connection may be long lasting and require a high constant throughput as well as providing robust, high-speed communications in wireless mesh networks deployed in rural environments, where there are significant amounts of spectrum available for secondary use. Better transmission scheduling can lead to improved network efficiency and less network resource consumption, e.g. energy-use. The problem is hard to due the presence of both intra-flow and inter-flow interference. In this paper, we develop a new polynomial time constant-factor approximation algorithm for this problem. We also present an effective heuristic method for finding effective multipath routes. It has been shown by simulation results that the end-to-end throughput given by the proposed algorithms provide nearly twice the throughput of single path routes and that the schedules generated are close to optimal.