We consider the problem of identifying a linear, time-invariant system from its noisy input/output data. The input and output are assumed to be non-Gaussian, while the input and output noises are assumed to be mutually correlated, colored, and Gaussian. Using third-order cross- and auto-cumulants, we extend the well-known Steiglitz-McBride identification method to cumulant domains, and show that it is consistent under a certain `third-order' persistency of excitation condition. By comparison, the Steiglitz-McBride method is not consistent when either input noise is present or when the output noise is colored. For an empirical assessment, we provide simulations that demonstrate the proposed method's usefulness.