Relativistic optics in moving media is an old topic in electromagnetic theory. The study of bianisotropic media – a very important class of metamaterials – stems from the research in that topic. However, a coordinate-free approach to moving media leads to rather cumbersome algebraic manipulations with tensors (or dyadics). In this article we present an alternative framework to deal with moving media: using spacetime algebra, the Clifford (geometric) algebra of Minkowski spacetime, we show that the inherent complexity of this subject can be substantially reduced through a transformation – the vacuum form reduction – that maps our problem into an equivalent spacetime with the same formal spacetime constitutive relation as real vacuum. This is made clear by showing that the two Maxwell equations (in spacetime) are reduced to a single Maxwell equation as in vacuum. We apply this technique to plane wave propagation in moving isotropic media which, from the laboratory perspective (that sees the movement), are actually nonreciprocal bianisotropic media. Our results are in full agreement with the conventional tensor and dyadic analyses.