In this paper we extend to the multidimensional case the modified Poisson series representation of linear stochastic processes driven by $\alpha$-stable innovations. The latter has been recently introduced in the literature and it involves a Gaussian approximation of the residuals of the series, via the exact characterization of their moments. This allows for Bayesian techniques for parameter or state inference that would not be available otherwise, due to the lack of a closed-form likelihood function for the $\alpha$-stable distribution. Simulation results are presented to validate the introduced extension and the quality of the approximation of the distribution. Finally, we show an example of generation from the process.