A number of different processing techniques have been developed to design and fabricate three-dimensional (3D) scaffolds for tissue-engineering applications. The imperfection of the current techniques has encouraged the use of a rapid prototyping technology known as fused deposition modeling (FDM). Our results show that FDM allows the design and fabrication of highly reproducible bioresorbable 3D scaffolds with a fully interconnected pore network. The mechanical properties and in vitro biocompatibility of polycaprolactone scaffolds with a porosity of 61 ± 1% and two matrix architectures were studied. The honeycomb-like pores had a size falling within the range of 360 × 430 × 620 μm. The scaffolds with a 0/60/120° lay-down pattern had a compressive stiffness and a 1% offset yield strength in air of 41.9 ± 3.5 and 3.1 ± 0.1 MPa, respectively, and a compressive stiffness and a 1% offset yield strength in simulated physiological conditions (a saline solution at 37 °C) of 29.4 ± 4.0 and 2.3 ± 0.2 MPa, respectively. In comparison, the scaffolds with a 0/72/144/36/108° lay-down pattern had a compressive stiffness and a 1% offset yield strength in air of 20.2 ± 1.7 and 2.4 ± 0.1 MPa, respectively, and a compressive stiffness and a 1% offset yield strength in simulated physiological conditions (a saline solution at 37 °C) of 21.5 ± 2.9 and 2.0 ± 0.2 MPa, respectively. Statistical analysis confirmed that the five-angle scaffolds had significantly lower stiffness and 1% offset yield strengths under compression loading than those with a three-angle pattern under both testing conditions (p ≤ 0.05). The obtained stress–strain curves for both scaffold architectures demonstrate the typical behavior of a honeycomb structure undergoing deformation. In vitro studies were conducted with primary human fibroblasts and periosteal cells. Light, environmental scanning electron, and confocal laser microscopy as well as immunohistochemistry showed cell proliferation and extracellular matrix production on the polycaprolactone surface in the 1st culturing week. Over a period of 3–4 weeks in a culture, the fully interconnected scaffold architecture was completely 3D-filled by cellular tissue. Our cell culture study shows that fibroblasts and osteoblast-like cells can proliferate, differentiate, and produce a cellular tissue in an entirely interconnected 3D polycaprolactone matrix.