Understanding the mechanosensitivity of tissues is a fundamentally important problem having far-reaching implications for tissue engineering. Here we study vascular networks formed by a coculture of fibroblasts and endothelial cells embedded in three-dimensional biomaterials experiencing external, physiologically relevant forces. We show that cyclic stretching of the biomaterial orients the newly formed network perpendicular to the stretching direction, independent of the geometric aspect ratio of the biomaterial's sample. A two-dimensional theory explains this observation in terms of the network's stored elastic energy if the cell-embedded biomaterial features a vanishing effective Poisson's ratio, which we directly verify. We further show that under a static stretch, vascular networks orient parallel to the stretching direction due to force-induced anisotropy of the biomaterial polymer network. Finally, static stretching followed by cyclic stretching reveals a competition between the two mechanosensitive mechanisms. These results demonstrate tissue-level mechanosensitivity and constitute an important step toward developing enhanced tissue repair capabilities using well-oriented vascular networks.