We provide effective methods to construct and manipulate trilinear birational maps $\phi:(\mathbb{P}^1)^3\dashrightarrow \mathbb{P}^3$ by establishing a novel connection between birationality and tensor rank. These yield four families of nonlinear birational transformations between 3D spaces that can be operated with enough flexibility for applications in computer-aided geometric design. More precisely, we describe the geometric constraints on the defining control points of the map that are necessary for birationality, and present constructions for such configurations. For adequately constrained control points, we prove that birationality is achieved if and only if a certain $2\times 2\times 2$ tensor has rank one. As a corollary, we prove that the locus of weights that ensure birationality is $\P^1\times\P^1\times\P^1$. Additionally, we provide formulas for the inverse $\phi^{-1}$ as well as the explicit defining equations of the irreducible components of the base loci. Finally, we introduce a notion of ``distance to birationality'' for trilinear rational maps, and explain how to continuously deform birational maps.