The multinomial probit model has emerged as a useful framework for modeling nominal categorical data, but extending such models to multivariate measures presents computational challenges. Following a Bayesian paradigm, we use a Markov chain Monte Carlo (MCMC) method to analyze multivariate nominal measures through multivariate multinomial probit models. As with a univariate version of the model, identification of model parameters requires restrictions on the covariance matrix of the latent variables that are introduced to define the probit specification. To sample the covariance matrix with restrictions within the MCMC procedure, we use a parameter-extended Metropolis-Hastings algorithm that incorporates artificial variance parameters to transform the problem into a set of simpler tasks including sampling an unrestricted covariance matrix. The parameter-extended algorithm also allows for flexible prior distributions on covariance matrices. The prior specification in the method described here generalizes earlier approaches to analyzing univariate nominal data, and the multivariate correlation structure in the method described here generalizes the autoregressive structure proposed in previous multiperiod multinomial probit models. Our methodology is illustrated through a simulated example and an application to a cancer-control study aiming to achieve early detection of breast cancer.