We perform simulations of obstructed diffusion in periodic lattice configurations of solid, nonoverlapping, impermeable superballs to study effective diffusivity as a function of shape and solid volume fraction for simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices. In the simulations, point particles diffuse in the intermediate space between the solid superballs. For sc lattices, we find that for a constant solid volume fraction, the optimal effective diffusivity is obtained for a particular superball between a sphere and a cube, whereas for bcc and fcc lattices, no such optimum exists. Further, we investigate the prediction of the effective diffusivity in these systems using an approximate analytical expression based on two- and three-point correlation functions and the microstructural parameter ζ2.