This paper develops a robust framework for the multiscale design of three-dimensional lattices with macroscopically tailored structural characteristics. The work exploits the high process flexibility and precision of additive manufacturing to the physical realization of complex microstructure of metamaterials by developing and implementing a multiscale approach. Structures derived from such metamaterials exhibit properties which differ from that of the constituent base material. A periodic microscale model is developed whose geometric parameterization enables smoothly changing properties and for which the connectivity of neighboring microstructures in the large-scale domain is guaranteed by slowly changing large-scale descriptions of the lattice parameters. A lattice-based functional grading of material is derived using the finite element method with sensitivities derived by the adjoint method. The novelty of the work lies in the use of multiple geometry-based small-scale design parameters for optimization problems in three-dimensional real space. The approach is demonstrated by solving a classical compliance minimization problem. The results show improved optimality compared to commonly implemented structural optimization algorithms.