In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, \emph{J. Comput. Appl. Math.}, extbf{270} (2014), 88–99].