This paper presents a novel approach to structure determination of linear systems along with the choice of system orders and parameters. AutoRegressive (AR), Moving Average (MA) or AutoRegressive-Moving Average (ARMA) model structure can be extracted blindly from the Third Order Cumulants (TOC) of the system output measurements, where the unknown system is driven by an unobservable stationary independent identically distributed (i.i.d.) non-Gaussian signal. By means of the system order recursion, whether the system has an AR structure or has AR part of an ARMA structure is firstly investigated. MA features in the TOC domain is then applied as a threshold to decide if the system is an MA model or has MA part of an ARMA model. Numerical simulations illustrate the generality of the proposed blind structure identification methodology that may serve as a guideline for blind linear system modeling.