Time-varying material responses create new opportunities for light technologies [1-7]. Time modulated materials potentially offer an unprecedented control of the electromagnetic radiation, as they enable nonreciprocal waveguiding and strongly nonreciprocal light-matter interactions, without an external magnetic bias. Recently, it was shown that spacetime crystals with a travelling wave-type modulation – such that the material parameters vary both in space and in time – allows for the synthesis of rather general effective non-reciprocal metamaterial responses in the long wavelength limit [4,5]. The homogenization approach relies on a Galilean coordinate transformation, which is used to switch to a co-moving frame where the material parameters become independent of time. The spacetime crystal response is homogenized in the co-moving frame, and then, the effective response is determined in the original laboratory frame by using an inverse Galilean transformation. Remarkably, a spacetime crystal formed by isotropic materials behaves effectively as a synthetic moving medium in the long wavelength limit. Interestingly, the velocity that determines the synthetic Fresnel drag is different from the modulation velocity v, both in amplitude and sign. Furthermore, recently we introduced an analytical formalism to homogenize spacetime crystals made of anisotropic crystals [6]. It was demonstrated that anisotropic spacetime crystals with a suitable glide-rotation symmetry behave effectively as Tellegen (axion) media [7-9]. This peculiar form of bianisotropic and nonreciprocal response is interesting not only for applications, but also to emulate the hypothetical response of axion particles which are relevant in high-energy physics. Inspired by these developments, here we present our most recent findings on the wave propagation in multidimensional spacetime crystals, i.e. systems periodic both in time and in two or more directions of space. For a travelling wave modulation, our analysis reveals a number of peculiar features such as: (ii) exotic nonreciprocal and bianisotropic responses in the long wavelength limit, (ii) we unveil new geometries where the multi-dimensional crystal behaves exactly as a moving medium; in particular, the synthetic Fresnel drag velocity coincides exactly with the modulation speed, (iii) a complex bandgap structure tailored by the relativistic Doppler effect.