Count data over spatial lattices are the building blocks of spatial econometric data (e.g. unemployment rates in small areas). We consider a hierarchical statistical model made up of a Poisson model for the counts and an underlying Spatial Random Effects process for the logarithm of the mean of the Poisson distribution. The resulting dimension reduction leads to substantial computational speed-ups. These models make no assumptions of homogeneity, stationarity, or isotropy. We develop maximum-likelihood estimates (MLEs) for the parameters of the underlying process using an EM algorithm, and we predict unknown mean counts over the entire spatial lattice.