New exact solutions for unsteady magnetohydrodynamic (MHD) flows of a generalized second-grade fluid near a nonuniform accelerating plate have been derived. The generalized second-grade fluid saturates the porous space. Fractional derivative is used in the governing equation. The analytical expressions for velocity and shear stress fields have been obtained by using Laplace transform technique for the fractional calculus. The obtained solutions are expressed in series form in terms of Fox H-functions. Similar solutions for ordinary second-grade fluid passing through a porous space are also recovered. Moreover, several figures are sketched for the pertinent parameters to analyze the characteristics of velocity field and shear stress.