This paper addresses the potential of self-made polyester-urethane filament as a candidate for Fused Filament Fabrication (FFF)-based 3D printing (3DP) in medical applications. Since the industry does not provide many ready-made solutions of medical-grade polyurethane filaments, we undertook research aimed at presenting the process of thermoplastic polyurethane (TPU) filament formation, detailed characteristics, and 3DP of specially designed elastic porous structures as candidates in cancellous tissue engineering. Additionally, we examined whether 3D printing affects the structure and thermal stability of the filament. According to the obtained results, the processing parameters leading to the formation of high-quality TPU filament (TPU_F) were captured. The results showed that TPU_F remains stable under the FFF 3DP conditions. The series of in vitro studies involving long- and short-term degradation (0.1 M phosphate-buffered saline (PBS); 5 M sodium hydroxide (NaOH)), cytotoxicity (ISO 10993:5) and bioactivity (simulated body fluid (SBF) incubation), showed that TPU printouts possessing degradability of long-term degradable tissue constructs, are biocompatible and susceptible to mineralization in terms of hydroxyapatite (HAp) formation during SBF exposure. The formation of HAp on the surface of the specially designed porous tissue structures (PTS) was confirmed by scanning electron microscope (SEM) and energy-dispersive X-ray spectroscopy (EDS) studies. The compression test of PTS showed that the samples were strengthened due to SBF exposure and deposited HAp on their surface. Moreover, the determined values of the tensile strength (~30 MPa), Young’s modulus (~0.2 GPa), and compression strength (~1.1 MPa) allowed pre-consideration of TPU_F for FFF 3DP of cancellous bone tissue structures.