The Ba-Ir-Ge ternary compounds BaIr2Ge7 and Ba3Ir4Ge16 exhibit superconductivity (SC) at 2.5 and 5.2 K, respectively. Detailed single-crystal structural analysis revealed that these compounds share unique quasi-two-dimensional networks composed of crown-shaped Ge rings that accommodate Ba atoms at the center, referred to as "edge-shared crown-shaped BaGe16 polyhedra". The layered Ba-Ge network yielded a modest anisotropy of 1.3-1.4 in the upper critical field, which is in good agreement with the band structure calculations. The Ba-Ge structural unit is similar to cage structures seen in various clathrates in which the anharmonic vibration of the central atoms, the so-called "rattling" behavior, brings about strong-coupling SC. However, each Ba-Ge unit is relatively small compared to these materials, which likely excludes the possibility of unconventional SC.