The structures of three quasicrystal approximant phases in the TM-Mg-Zn (TM = Hf, Zr, Ti) systems with the analyzed compositions Hf5Mg18Zn77, Zr5Mg18Zn77, and Ti5.5Mg17.5Zn77 have been synthesized, and their structures have been analyzed by single-crystal X-ray diffraction. The structure analyses revealed that these cubic phases with the space group Pm contain two different rhombic-triacontahedral clusters. These clusters are so-called Bergman-type atomic clusters and previously known approximants of face-centered icosahedral (F-type) quasicrystals are composed only of Mackay-type clusters, thus these compounds are seen as new prototype structures. Electron density maps calculated by the maximum entropy method (MEM) show that one of the atomic clusters displays characteristic structural disorder. The disorder in these phases is related to the chemical intergrowth of different Friauf polyhedra, and the prospects of new guide lines for finding quasicrystals composed of such polyhedra are discussed.