These amplifiers and filters rely on optomechanically induced transparency (OMIT) and absorption (OMIA), respectively. Such devices can amplify microwave signals with large, controllable gain, high dynamic range and very low noise. Furthermore, extremely narrowband filters can be constructed with this technique. We briefly review previous measurements of microwave OMIT and OMIA before reporting our own measurements of these phenomena, which cover a larger parameter space than has been explored in previous works. We find excellent agreement between our measurements and the predictions of input/output theory, thereby guiding further development of microwave devices based on nanomechanics.