This work analyzes constrained black box optimization in which the functions defining the problem are periodic with respect to some or all the variables. We show that the natural strategy of mapping trial points into the interval defined by the period in the Mesh Adaptive Direct Search (MADS) framework can be easily done in practice, and the convergence analysis does not suffer any loss if some minor algorithmic conditions are met. The proposed strategy is tested on a nonsmooth classification problem and on a bi-objective portfolio selection problem for which MADS is repeatedly used on mono-objective problems.