We study the optimization of two-dimensional bodies featuring a locally periodic microstructure. We consider a linearly elastic material with holes and minimize the compliance of the macroscopic body. We implement a three stage process. In the first stage, the holes are considered to be infinitesimal and the theory of homogenization is used to describe the elastic properties of the macroscopic body. The periodicity lattice varies from point to point at a macroscopic scale; these variations are described by a ``grid map'' transporting a regular square lattice into a deformed (non-rectangular) lattice. At this stage, we seek for an optimal distribution of the homogenized material. The main novelty of this paper is that the optimization variable is the grid map itself, together with three other scalar functions describing the geometry of the microscopic holes. Another difference relatively to previous works is that we allow for skew and/or elongated periodicity cells. After the optimization has reached convergence, there is a second stage where this grid map is used for transforming the optimized homogenized structures into manufacturable designs. These are curved grillages featuring small but macroscopic holes. Since the grid map is computed in the first stage, the second stage requires almost zero computational effort. In a third stage of the process, unnecessary features (bars carrying no loads) are removed by a pure shape optimization algorithm on a fine mesh. This stage is computationally heavy but only a few steps are needed because the starting point (the curved grillage obtained in the second stage) is already nearly optimal.