In this paper, an efficient technique for computing the bound state energies and wave functions of the Schrödinger Equation (SE) associated with a new class of spherically symmetric hyperbolic potentials is developed. This technique is based on a recent approximation scheme for the orbital centrifugal term and on the use of the Fröbenius method (FM). The bound state eigenvalues are given as zeros of calculable functions. The corresponding eigenfunctions can be obtained by substituting the calculated energies into the recurrence relations for the expanding coefficients of the Fröbenius series representing the solution. The excellent performance of this technique is illustrated through numerical results for some special cases like Pöschl-Teller potential (PTP), Manning-Rosen potential (MRP) and Pöschl-Teller polynomial potential (PTPP), with an application to the Gaussian potential well (GPW). Comparison with other methods is presented. Our results agree noticeably with the previously reported ones.