We propose a new fiducial Markov Chain Monte Carlo (MCMC) method for fitting parametric Gaussian models. We utilize the Cayley transform to decompose the parametric covariance matrix, which in turn allows us to formulate a general data generating algorithm for Gaussian data. Leveraging constrained generalized fiducial inference, we are able to create the basis of an MCMC algorithm, which can be specified to parametric models with minimal effort. The appeal of this novel approach is the wide class of models which it permits, ease of implementation and the posterior-like fiducial distribution without the need for a prior. We provide background information for the derivation of the relevant fiducial quantities, and a proof that the proposed MCMC algorithm targets the correct fiducial distribution. We need not assume independence nor identical distribution of the data, which makes the method attractive for application to time series and spatial data. Well-performing simulation results of the MA(1) and Matérn models are presented.