In robotics, the origami-based design methodology uses straightforward fabrication and assembly processes to create small-scale parallel mechanisms. Delta mechanisms are among the well-known parallel mechanisms used mostly in pick-and-place operations due to their capability to reach high speeds and accelerations. In this work, we present a novel Delta mechanism based on origami-inspired designs and two-dimensional layer-by-layer fabrication methods, reducing the time and errors in manufacturing. We developed a new flat parallelogram providing translations in X-Y-Z directions, respecting the Delta mechanism's conventional kinematic models. The fabrication and assembly processes include laser machining and lamination, eliminating manual folding and bonding steps. The mechanism operates in a 20 x 20 x 20 mm(3) workspace and a 17.5 cm diameter circular footprint when it is entirely flat. The kinematic performance of the mechanism is analyzed using a six degrees-of-freedom position sensor on the end effector. The experiments are conducted to follow circular trajectories with varying radii at different heights. Despite having no feedback control from the end effector, the mechanism follows the given trajectories with 1.5 mm root-mean-square (RMS) precision. We also present the effects of the elasticity of flexible materials at different regions of the mechanism on the performance of the Delta robot.