Block matrices associated with discrete Trigonometric transforms (DTT's) arise in the mathematical modelling of several applications of wave propagation theory including discretizations of scatterers and radiators with the Method of Moments, the Boundary Element Method, and the Method of Auxiliary Sources. The DTT's are represented by the Fourier, Hartley, Cosine, and Sine matrices, which are unitary and offer simultaneous diagonalizations of specific matrix algebras. The main tool for the investigation of the aforementioned wave applications is the efficient inversion of such types of block matrices. To this direction, in this paper we develop an efficient algorithm for the inversion of matrices with ${U}$-diagonalizable blocks (${U}$ a fixed unitary matrix) by utilizing the ${U}$-diagonalization of each block and subsequently a similarity transformation procedure. We determine the developed method's computational complexity and point out its high efficiency compared to standard inversion techniques. An implementation of the algorithm in Matlab is given. Several numerical results are presented demonstrating the CPU-time efficiency and accuracy for ill-conditioned matrices of the method. The investigated matrices stem from real-world wave propagation applications.