In this work attention is directed to general structural optimization problems considering discrete-continuous design variables. The optimization problem is formulated as the minimization of an objective function subject to multiple design requirements. The mathematical programming statement is set into the framework of a Bayesian model updating problem. Constraints are handled directly within the proposed scheme, generating designs distributed over the feasible design space. Based on these samples, a set of designs lying in the vicinity of the optimal solution set is obtained. The Bayesian model updating problem is solved by an effective Markov chain Monte Carlo simulation scheme, where appropriate proposal distributions are introduced for the continuous and discrete design variables. The approach can efficiently estimate the sensitivity of the final design and constraints with respect to the design variables. In addition, the numerical implementation of the optimization algorithm depends on few control parameters. For illustration purposes, the general formulation is applied to an important class of problems, specifically, reliability-based design optimization of structural systems under stochastic excitation. Three numerical examples showing the effectiveness and potentiality of the approach reported herein are presented.