A wall structure (32, 42, 68, 70, 80) with layers (A, B, C, D, E) of non-random voids (26A, 26B, 28B, 30B) that interconnect to form discretely defined tortuous passages between an interior (21) and an exterior surface (23) of the wall for transpiration cooling of the wall. A coolant flow (38) through the wall may be metered by restrictions in coolant outlets (31) and/or within the passages to minimize the coolant requirement. Pockets (44) may be formed on the exterior surface of the wall for thermal Insulation (46). The layers may be formed by lamination, additive manufacturing, or casting. Layer geometries include alternating layers (A, B, C) with different overlapping void patterns (42), 3-D lattice structures (70), and offset waffle structures (80).