A critical feature of state-of-the-art microfluidic technologies is the ability to fabricate multilayer structures without relying on the expensive equipment and facilities required by soft lithography-defined processes. Here, three-dimensional (3D) printed polymer molds are used to construct multilayer poly(dimethylsiloxane) (PDMS) devices by employing unique molding, bonding, alignment, and rapid assembly processes. Specifically, a novel single-layer, two-sided molding method is developed to realize two channel levels, non-planar membranes/valves, vertical interconnects (vias) between channel levels, and integrated inlet/outlet ports for fast linkages to external fluidic systems. As a demonstration, a single-layer membrane microvalve is constructed and tested by applying various gate pressures under parametric variation of source pressure, illustrating a high degree of flow rate control. In addition, multilayer structures are fabricated through an intralayer bonding procedure that uses custom 3D-printed stamps to selectively apply uncured liquid PDMS adhesive only to bonding interfaces without clogging fluidic channels. Using integrated alignment marks to accurately position both stamps and individual layers, this technique is demonstrated by rapidly assembling a six-layer microfluidic device. By combining the versatility of 3D printing while retaining the favorable mechanical and biological properties of PDMS, this work can potentially open up a new class of manufacturing techniques for multilayer microfluidic systems.