For materials used in high-temperature thermoelectric power generation, the choices are still quite limited. Here we demonstrate the design and synthesis of a new class of complex Zintl compounds, Ca1-xRExAg1-ySb (RE = La, Ce, Pr, Nd, Sm) (P6(3)mc, No. 186, LiGaGe-type), which exhibit a high figure of merit in the high-temperature region. Compared with the parent structure that is based on CaAgSb (Pnma, No. 62, TiNiSi-type), an interesting structural relationship is established which suggests that important size and electronic effects govern the formation of these multinary phases. According to theoretical calculations, such a structural transformation from the orthorhombic TiNiSi-type to the hexagonal LiGaGe-type also corresponds to an obvious modification in the electronic band structure, which explains the observed significant enhancement of the related thermoelectric properties. For an optimized p-type material, Ca0.84Ce0.16Ag0.87Sb, a figure of merit of similar to 0.7 can be achieved at 1079 K, which is comparable to that of Yb14MnSb11 at the same temperature. In addition, due to the excellent thermal stability and high electrical conductivity, these materials are very promising candidates for high-temperature thermoelectric power generation.