Strong enhancement of second-harmonic generation (SHG) is expected in one-dimensional microcavities when double resonance for the pump and the harmonic fields, as well as phase matching, are achieved. The realization of a doubly resonant microcavity with dielectric mirrors made of nonbirefringent materials is difficult because of the refractive index dispersion of the constituent media. Here we present a powerful method, based on photonic crystal concepts like gap maps and their generalization to defect modes, for the design of doubly resonant microcavities with periodic dielectric mirrors. The material dispersion is compensated by using the angle of incidence as a tuning parameter, thanks to the polarization splitting of cavity modes. The cavity enhancement of SHG increases exponentially with the number of periods in the dielectric mirrors and can be much larger than in single-resonant microcavities with comparable (or even larger) quality factors. The roles of phase delay and of thin versus thick configurations in the dielectric mirrors, of the growth orientation, and of the polarization degrees of freedom in achieving double resonance with phase matching are discussed. Significant examples of doubly resonant SHG with high conversion efficiency are given for Al0.25Ga0.75As cavities with Al0.4Ga0.6As/Alox (oxidized AlAs) mirrors.