For general non-Gaussian distributions, the covariance and precision matrices do not encode the independence structure of the variables, as they do for the multivariate Gaussian. This paper builds on previous work to show that for a class of non-Gaussian distributions -- those derived from diagonal transformations of a Gaussian -- information about the conditional independence structure can still be inferred from the precision matrix, provided the data meet certain criteria, analogous to the Gaussian case. We call such transformations of the Gaussian as the generalized nonparanormal. The functions that define these transformations are, in a broad sense, arbitrary. We also provide a simple and computationally efficient algorithm that leverages this theory to recover conditional independence structure from the generalized nonparanormal data. The effectiveness of the proposed algorithm is demonstrated via synthetic experiments and applications to real-world data.