We report a new type of three-dimensional architected polymer foams composed of perforated spherical shells and flat strut connectors, which can be precisely produced by 3D printing techniques. We investigate the effects of foam architectures, manufacturing process, and constitutive material on the deformation patterns and failure modes of the proposed architected foams. We demonstrate that flat strut connectors offer unprecedented design flexibility for controlling the mechanical performance. By tuning the geometric parameter of flat strut connectors, the stiffness of architected foams can increase about one order of magnitude while the relative density increases only by 5%. Furthermore, the failure modes can be engineered from a catastrophic one to a progressive one by using weak flat strut connectors. Our experiments elucidate the salient roles of the layer-by-layer manufacturing process and constitutive polymer on the mechanical behavior of the proposed architected foams.