It has been estimated that about 30% of the genes in the human genome are regulated by microRNAs (miRNAs). These are short RNA sequences that can down-regulate the levels of mRNAs or proteins in animals and plants. Genes regulated by miRNAs are called targets. Typically, methods for target prediction are based solely on sequence data and on the structure information. In this paper we propose a Bayesian graphical modeling approach that infers the miRNA regulatory network by integrating expression levels of miRNAs with their potential mRNA targets and, via the prior probability model, with their sequence/structure information. We use a directed graphical model with a particular structure adapted to our data based on biological considerations. We then achieve network inference using stochastic search methods for variable selection that allow us to explore the huge model space via MCMC. A time-dependent coefficients model is also implemented. We consider experimental data from a study on a very well-known developmental toxicant causing neural tube defects, hyperthermia. Some of the pairs of target gene and miRNA we identify seem very plausible and warrant future investigation. Our proposed method is general and can be easily applied to other types of network inference by integrating multiple data sources.