We present rigorous optimization and design of three-dimensional photonic-crystal (PhC) structures involving finite dielectric rods. These types of PhCs are known to be useful in diverse applications, such as imaging, power focusing, filtering, and pattern shaping at optical frequencies. Without resorting to their two-dimensional models, which are commonly used in the literature, we consider PhCs as three-dimensional structures, whose electromagnetic characteristics are optimized via genetic algorithms integrated with an efficient and accurate solver based on the multilevel fast multipole algorithm. In addition to the designs of PhCs that can provide desired output patterns, we present sensitivity tests to understand the critical geometric parameters, as well as to test and evaluate the tolerance of the proposed designs to possible fabrication errors.