We propose the concept of organic membrane photonic integrated circuits (OMPICs), which incorporate various functions needed for optical signal processing into a flexible organic membrane. We describe the structure of several devices used within the proposed OMPICs (e.g., transmission lines, I/O couplers, phase shifters, photodetectors, modulators), and theoretically investigate their characteristics. We then present a method of fabricating the photonic devices monolithically in an organic membrane and demonstrate the operation of transmission lines and I/O couplers, the most basic elements of OMPICs.