A rigorous framework and an efficient numerical implementation for computing leaky-wave poles for grounded inhomogeneous dielectric (multi-) layers are presented. The pertinent transverse-resonance (eigenvalue) equations, obtained by power-series solution of the related Sturm-Liouville problems, are solved by means of a Padé-approximant-based root-finding procedure. Numerical simulations show that very accurate results (ten decimal figures, on average) are usually obtained with a mild computational and programming effort, over a reasonably wide spectral range.