Here, we show that spacetime modulations provide an exciting route to engineer arbitrary metamaterial responses, namely, different classes of media with nonreciprocal couplings. We develop an analytical formalism to homogenize anisotropic spacetime crystals in the long wavelength limit. It is found that spacetime crystals with suitable glide-rotation symmetry can have a giant nonreciprocal response, several orders of magnitude larger than in natural materials. Synthetic nonreciprocal responses may have interesting applications in optics, for example in electromagnetic isolation, and in addition may enable exciting new forms of light-wave interactions.