In regenerative medicine, an engineered tissue is a composition of a sample of cells cultured on a spatially controlled medical device, called a biological scaffold (or just a bioscaffold). These devices are made of tissue-equivalent materials and represent the biological, mechanical, and spatial conditions in a specific type of human or animal tissue, becoming a possible way to replace damaged structures and develop artificial tissues or organs. Scaffolds with narrowly controlled characteristics—biological, mechanical and spatial properties—are vital for experiments mimicking in vivo conditions and in tissue regeneration scenarios. The aim of this narrative review is to identify and discuss the most important properties of these artificial constructs and the ways to achieve them via 3D printing-based technologies. Properties that can direct the development and differentiation of the cultured cells in a specific direction and ensure their biocompatibility and bioresorption, mechanical properties, spatial architecture, and porosity are discussed. The most common considerations in terms of the role of material selection, additives, and signal molecules and the appropriate spatially controlled manufacturing technologies for their assembly are covered, as are the radiological, biomechanical, and histological methods for their analysis. Finally, this paper highlights the challenges to the achievement of optimal scaffolds through additive manufacturing and gives suggestions for further research and development in this field.