The construction of birational maps is useful for several applications in geometric modeling, including the deformation of shapes by the free-form deformation technique. In this paper, we present effective methods to manipulate 3D birational tensor-product maps with entries of degree 1 × 1 × 1, which are the first for the construction of non-affine birational maps in 3D. To achieve birationality, specific geometric constraints on the Bézier control points are necessary as they permit the existence of certain syzygies. We provide criteria for the computation of suitable weights for the control points that yield birationality, and give explicit formulas for the inverse rational map. According to the degree of the inverse, we find different classes of birational maps. The first is the class of hexahedral maps, i.e. tensor-product maps whose control net determines the vertices of a quadrilaterally-faced hexahedron. Hexahedral birational maps are the direct generalization to 3D of 2D quadrilateral birational maps. In the remaining classes, the inverse rational map is quadratic for one of the parameters. Interestingly, the geometry of the control points is less constrained and the subsequent tensor-product map becomes more flexible.