The discretization of many continuous models can lead to large-scale optimization problems. As a means to accelerate the process of finding a solution, several multilevel procedures have been investigated, which exploit the solution of smaller problems corresponding to coarser discretization parameters. We consider directional direct-search methods endowed with polling strategies of Jacobi or Gauss-Seidel type along the coordinate directions. While it is well-known that such methods are generally unsuited to solving large-scale problems, we show that they can be dramatically accelerated when embedded in a derivative-free multilevel framework. We discuss some implementation issues, and present experiments on several test problems. We argue that our algorithms obtain competitive performance in practice, and that traditional limitations on the size of the problems tractable by classical direct-search methods can be overcome.