Recent work has highlighted how the phenomenon of flexoelectricity can masquerade as piezoelectricity. This notion can not only be exploited to create artificial piezoelectric-like materials without using piezoelectric materials, but may also explain measurement artifacts in dielectrics. In this article, we show that the reverse is also possible and potentially advantageous in certain situations (such as energy harvesting). By constructing a computational homogenization approach predicated on the finite element method, we argue that composites made of piezoelectric phases can conspire to endow the material with a distinct overall flexoelectric-like response even though the native flexoelectricity of the constituent materials is negligible. Full finite element procedures for numerical evaluation of the different effective tensors, including the flexoelectric tensor, are provided. Numerical investigations are conducted, showing variation of the effective flexoelectric properties with respect to local geometry and properties of the composite in piezoelectric-piezoelectric and polymer-piezoelectric composites. We find that the flexoelectric response can be tuned to nearly five times higher than the constituents.