We compute the dynamical polarization function for a graphene antidot lattice in the random-phase approximation. The computed polarization functions display a much more complicated structure than what is found for pristine graphene (even when evaluated beyond the Dirac-cone approximation); this reflects the miniband structure and the associated van Hove singularities of the antidot lattice. The polarization functions depend on the azimuthal angle of the q vector. We develop approximations to ease the numerical work and critically evaluate the performance of the various schemes. We also compute the plasmon dispersion law and find an approximate square-root dependence with a suppressed plasmon frequency as compared to doped graphene. The plasmon dispersion is nearly isotropic and the developed approximation schemes agree well with the full calculation.