Networks of Rydberg atoms provide a powerful basis for quantum simulators and quantum technologies. Inspired by matter-wave atomtronics, here we engineer switches, diodes, and universal logic gates composed of Rydberg networks. Our schemes control the dynamics of a Rydberg excitation via the antiblockade or facilitation mechanism, allowing for much faster devices compared to cold atom systems. Our approach is robust to noise and can be applied to individually trapped atoms and extensive three-dimensional gases. In analogy to electronics, Rydberg atomtronic devices promise to enhance quantum information processors and quantum simulators.