We describe a fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept. We use the interpolation data to implicitly define a convenient non-Hermitian structured matrix, and then apply a computationally efficient procedure for its triangular factorization. This leads to a transmission-line interpretation that makes evident the interpolation properties. We further discuss connections with the Lagrange interpolating polynomial as well as questions regarding the minimality and the admissible degrees of complexity of the solutions.