Three-dimensional bioprinting is the process of manipulating cell-laden bioinks to fabricate living structures. Three-dimensional bioprinting techniques have brought considerable innovation in biomedicine, especially in the field of tissue engineering, allowing the production of 3D organ and tissue models for in vivo transplantation purposes or for in-depth and precise in vitro analyses. Naturally derived hydrogels, especially those obtained from the decellularization of biological tissues, are promising bioinks for 3D printing purposes, as they present the best biocompatibility characteristics. Despite this, many natural hydrogels do not possess the necessary mechanical properties to allow a simple and immediate application in the 3D printing process. In this review, we focus on the bioactive and mechanical characteristics that natural hydrogels may possess to allow efficient production of organs and tissues for biomedical applications, emphasizing the reinforcement techniques to improve their biomechanical properties.