Three-dimensional (3D) extrusion printing of cellular/acellular structures with biocompatible materials has been widely investigated in recent years. However, the requirement of a suitable solidification rate of printable ink materials constrains the utilization of extrusion-based 3D printing techniques. In this study, the nanoclay yield-stress suspension-enabled extrusion-based 3D printing system has been investigated and demonstrated to overcome solidification rate constraints during printing. Utilizing the liquid–solid transition property of nanoclay suspension, two fabrication approaches, including nanoclay support bath-enabled printing and nanoclay-enabled direct printing, have been proposed. For the former approach, nanoclay (Laponite® EP) has been used as a support bath material to fabricate alginate-based tympanic membrane patches. The constituents of alginate-based ink have been investigated to have the desired mechanical property of alginate-based tympanic membrane patches and facilitate the printing process. For the latter approach, nanoclay (Laponite® XLG) has been used as an internal scaffold material to help print poly (ethylene glycol) diacrylate (PEGDA)-based neural chambers, which can be further cross-linked in air. Mechanical stress analysis has been performed to explore the geometric limitation of printable Laponite® XLG-PEGDA neural chambers.