Singe crystals of four new AIMII[ReO4]3 perrhenates (AI = Na, MII = Pb; AI = Na, MII = Sr, Ag, K) have been prepared from the respective AI[ReO4] and MII[ReO4]2 perrhenates using melt and solution evaporation techniques. All new compounds belong to the hexagonal CdTh[MoO4]3 structure type, similar to previously known NaCa[ReO4]3. We discuss the crystal chemical relationships within this structure type and suggest existence of some more representatives involving other tetrahedral anions.