The study of elastic wave propagation is a fundamental tool in different fields, from non-destructive damage evaluation to ultrasonic imaging. Usually these techniques rely on inversion methods based on homogenised theories, that are valid only when the wavelength of the perturbation is considerably larger than the characteristic size of the heterogeneities of the materials. Heterogeneities can occur at different scales, and then it is useful to fix the scales of interest: the micro scale (i.e. the scale of the constitutive material ~um), the meso scale (the scale of the architecture ~mm) and the macro scale (the scale of the tissue ~cm). When the wavelength approaches the characteristic size of the architecture, an upscaling occurs and mesoscopic effects can be transferred to macro-scale. In this case, classic models used in the aforementioned inversion procedures can fail to predict a the correct response and they need to be improved. In this work, we will address the case of non-centrosymmetric architectures, i.e. those for which the unit cell does not have any centre of inversion, and it will be shown that the effects on wave propagation in terms of dispersion and polarisation cannot be negliged in common applications involving elastic waves. We will also show that, in order to describe these media with a continuum model, the use of an enriched continuum theory, such as the strain gradient elasticity, is mandatory. The particular example of the gyroid unit cell is detailed.