The overall performance of multi-hop cognitive radio networks (MHCRNs) can be improved significantly by employing the diversity of orthogonal licensed channels in underlay fashion. However, the mutual interference between secondary links and primary links and the congestion due to the contention among traffic flows traversing the shared link become obstacles to this realizing technique. How to control congestion efficiently in coordination with power and spectrum allocation optimally in order to obtain a high end-to-end throughput is motivating cross-layer designs for MHCRNs. In this paper, by taking into account the problem of joint rate adaption, power control, and spectrum allocation (JRPS), we propose a new cross-layer optimization framework for MHCRNs using orthogonal frequency division multiple access (OFDMA). Specifically, the JRPS formulation is shown to be a mix-integer non-linear programming (MINLP) problem, which is NP-Hard in general. To solve the problem, we first develop a partially distributed algorithm, which is shown to converge to the global optimum within a reasonable time interval. We next propose a suboptimal solution which addresses the shortcomings of the first. Using numerical results, we finally demonstrate the efficiency of the proposed algorithms.