We report a giant Voigt effect in diluted magnetic semiconductors ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Te and ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Se and present an excitonic model that describes its dispersion and ${\mathit{M}}^{2}$ dependence, M being the magnetization. We discuss the Voigt effect in ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Te (0.10x0.45), a zinc-blende diluted magnetic semiconductor, in the context of this model. In addition, we present experimental results for the Voigt effect in ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Se (x=0.26 and 0.31), a diluted magnetic semiconductor with wurtzite structure, where the effect is considerably more complicated due to its lower symmetry. Specifically, we present a striking anisotropy of the Voigt effect in ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Se, which depends on the relative orientation of the applied magnetic field with respect to the optic (c^) axis of the crystal.