The recently proposed concept of metamaterials has opened exciting venues to control wave-matter interactions in unprecedented ways. Here, we demonstrate the relevance of metamaterials for inducing acoustic birefringence, a phenomenon which has already found its versatile applications in optics in designing light modulators or filters, and nonlinear optic components. This is achieved in a suitably designed acoustic metamaterial which is non-Eulerian, in the sense that at low frequencies, it cannot be homogenized to a uniform acoustic medium whose behavior is characterized by the Euler equation. Thanks to the feasibility of engineering its subwavelength structure, such a non-Eulerian metamaterial allows one to desirably manipulate the birefringence process. Our findings may give rise to the generation of novel devices such as tunable acoustic splitters and filters.