A very accurate method is presented for the continued-fraction expansion of the electron Green's-function propagator in crystals. The novelty consists in implementing the recursion technique in decoupled subzones of the first Brillouin zone, rather than in the whole Brillouin zone or in real space. With this procedure a very large number (of the order of a hundred or more) of continued-fraction steps becomes easily accessible. Our resummation technique is applied to a number of elemental and III-V semiconductors; in particular, results for silicon, germanium, gallium arsenide, and aluminum phosphide are presented as an example. We also discuss the problem of asymptotic behavior of continued-fraction coefficients.