The recent advances in material science and engineering disciplines have had a significant impact on the robotics field where numerous bioinspired soft robots have been developed. Roboticists can now develop and fabricate soft robotic systems made of materials with low elasticity using additive manufacturing technologies. One of the most studied classes of soft robotic systems is soft adaptive grippers. In this work, we present a novel fully 3D printed soft pneumatic gripper that incorporates a novel design of soft pneumatic chambers and a bioinspired fin-ray structure for conformal grasping. The soft gripper was printed in a single step using a low-cost and open-source fused deposition modeling (FDM) 3D printer and a commercially available thermoplastic poly(urethane) (TPU). A soft pneumatic actuator representing each finger of the gripper was optimized using finite element modeling (FEM). The FEM simulations predicted accurately the performance of the actuator in terms of deformation and blocked force. The high conformability of the proposed soft gripper was validated experimentally. The soft gripper was demonstrated lifting a heavy load and grasping a wide variety of objects with different weights, shapes, textures and stiffnesses.