The Radon transform and its inverse are important tools in producing images from medical devices such as CT (Computerized Tomography) scanners. Chebyshev spectral methods are an accurate and efficient tool for numerical approximating solutions to partial differential equations. The goal of this work is to develop a novel class of numerical methods that combine the Radon transform and Chebyshev spectral methods to produce an accurate and efficient framework for solving multidimensional linear hyperbolic partial differential equations (PDEs). The key innovation is that the Radon transform allows us to only solve a set of decoupled one-dimensional PDEs along each Radon transform direction, while the multidimensional nature of the solution is recovered in the Radon inversion process. This work aims to develop efficient algorithms for computing the Radon transform, computing the solution to the resulting set of PDEs, and computing the inverse Radon transform.