Prompted by the recent successful predictions of the internal dynamics of Andromeda's satellite galaxies, we revisit the classical Milky Way dwarf spheroidal satellites Draco, Sculptor, Sextans, Carina and Fornax in the framework of Milgromian dynamics (MOND). We use for the first time a Poisson solver with adaptive mesh refinement (AMR) in order to account simultaneously for the gravitational influence of the Milky Way and its satellites. This allows us to rigorously model the important external field effect (EFE) of Milgromian dynamics, which can reduce the effective acceleration significantly. We make predictions on the dynamical mass-to-light ratio (Mdyn/L) expected to be measured by an observer who assumes Newtonian dynamics to be valid. We show that Milgromian dynamics predicts typical Mdyn/L ≈ 10-50 M⊙/L⊙. The results for the most luminous ones, Fornax and Sculptor, agree well with available velocity dispersion data. Moreover, the central power-law slopes of the dynamical masses agree exceedingly well with values inferred observationally from velocity dispersion measurements. The results for Sextans, Carina and Draco are low compared to usually quoted observational estimates, as already pointed out by Angus. For Milgromian dynamics to survive further observational tests in these objects, one would thus need that either (a) previous observational findings based on velocity dispersion measurements have overestimated the dynamical mass due to, e.g. binaries and contaminant outliers, (b) the satellites are not in virial equilibrium due to the Milky Way tidal field, or (c) the specific theory used here does not describe the EFE correctly (e.g. the EFE could be practically negligible in some other theories), or a combination of (a)–(c).