The still-growing field of additive manufacturing (AM), which includes 3D printing, has enabled manufacturing of patient-specific medical devices with high geometrical accuracy in a relatively quick manner. However, the development of materials with specific properties is still ongoing, including those for enhanced bone-repair applications. Such applications seek materials with tailored mechanical properties close to bone tissue and, importantly, that can serve as temporary supports, allowing for new bone ingrowth while the material is resorbed. Thus, controlling the resorption rate of materials for bone applications can support bone healing by balancing new tissue formation and implant resorption. In this regard, this work aimed to study the combination of polylactic acid (PLA), polycaprolactone (PCL) and hydroxyapatite (HA) to develop customized biocompatible and bioresorbable polymer-based composite filaments, through extrusion, for fused filament fabrication (FFF) printing. PLA and PCL were used as supporting polymer matrices while HA was added to enhance the biological activity. The materials were characterized in terms of mechanical properties, thermal stability, chemical composition and morphology. An accelerated degradation study was executed to investigate the impact of degradation on the above-mentioned properties. The results showed that the materials’ chemical compositions were not affected by the extrusion nor the printing process. All materials exhibited higher mechanical properties than human trabecular bone, even after degradation with a mass loss of around 30% for the polymer blends and 60% for the composites. It was also apparent that the mineral accelerated the polymer degradation significantly, which can be advantageous for a faster healing time, where support is required only for a shorter time period.