The extraction of information from measured data about the interactions taking place in a network of systems is a key topic in modern applied sciences. This topic has been traditionally addressed considering bivariate time series, providing methods that are sometimes difficult to extend to multivariate data, the limiting factor being the computational complexity. Here, we present a computationally viable method based on black-box modeling which, whilst theoretically applicable only when a deterministic hypothesis about the processes behind the recordings is plausible, proves to work also when this assumption is severely affected. Conceptually, the method is very simple and is composed of three independent steps: in the first step a state-space reconstruction is performed separately on each measured signal; in the second step, a local model, i.e. a nonlinear dynamical system, is fitted separately on each (reconstructed) measured signal; afterwards, a linear model of the dynamical interactions is obtained cross-relating the (reconstructed) measured variables to the dynamics unexplained by the local models. The method is successfully validated on numerically generated data. An assessment of its sensitivity to data length and modeling and measurement noise intensity, and of its applicability to large scale systems, is also provided.