In this paper, we develop a new algorithmic framework to solve black-box problems with integer variables. The strategy included in the framework makes use of specific search directions (so called primitive directions) and a suitably developed nonmonotone line search, thus guaranteeing a high level of freedom when exploring the integer lattice. First, we describe and analyze a version of the algorithm that tackles problems with only bound constraints on the variables. Then, we combine it with a penalty approach in order to solve problems with simulation constraints. In both cases we prove finite convergence to a suitably defined local minimum of the problem. We report extensive numerical experiments based on a test bed of both bound-constrained and generally-constrained problems. We show the effectiveness of the method when compared to other state-of-the-art solvers for black-box integer optimization.