Reported are two new series of Zintl phases, ACd(4)Pn(3) and AZn(4)Pn(3) (A = Na, K, Rb, Cs; Pn = As, P), whose structures feature complex atomic arrangements based on four- and eight-coordinated arsenic and phosphorus. A total of 12 compounds have been synthesized from the corresponding elements via high temperature reactions, and their structures have been established by X-ray diffraction. RbCd4As3, KCd4As3, NaCd4As3, NaZn4As3, KCd4P3, and KZn4P3 crystallize with a new rhombohedral structure (space group R (3) over barm, Z = 3, Pearson symbol hR24), while the isoelectronic RbZn4As3, CsCd4As3, CsZn4As3, KZN(4)As(3), CsZn4P3, and RbZn4P3 adopt the tetragonal KCu4S3-type structure (space group P4/mmm, Z = 1, Pearson symbol tP8). Both structures are very closely related to the; ubiquitous CaAl2Si2 and ThCr2Si2 structure types, and the corresponding relationships are discussed. The experimental results have been complemented by linear muffin-tin orbital (LMTO) tight-binding band structure calculations. Preliminary transport properties measurements on polycrystalline samples suggest that the compounds of these families could be promising thermoelectric materials.