We describe a novel stochastic search algorithm for rapidly identifying regions of high posterior probability in the space of decomposable, graphical and hierarchical log-linear models. Our approach is based on the conjugate priors for log-linear parameters introduced in Massam, Liu and Dobra, 2008. We discuss the computation of Bayes factors through Laplace approximations and the Bayesian Iterate Proportional Fitting algorithm for sampling model parameters. We also present a clustering algorithm for discrete data. We compare our model determination approach with similar results based on multivariate normal priors for log-linear models. The examples concern a six-way, an eight-way and a sparse sixteen-way contingency tables. Extensions of this work involve stochastic algorithms for variable selection in regressions with discrete data. Relevant applications include building classifiers from gene expression, SNP and phenotype data.