Cold atom quantum sensors based on atom interferometry are among the most accurate instruments used in fundamental physics, metrology, and foreseen for autonomous inertial navigation. However, they typically have optically complex, cumbersome, and low-bandwidth atom detection systems, limiting their practical applications. Here, we demonstrate an enabling technology for high-bandwidth, compact, and nondestructive detection of cold atoms, using microwave radiation. We measure the reflected microwave signal to coherently and distinctly detect the population of single quantum states with a bandwidth close to 30 kHz and a design destructivity that we set to 0.04%. We use a horn antenna and free-falling molasses cooled atoms in order to demonstrate the feasibility of this technique in conventional cold atom interferometers. This technology, combined with coplanar waveguides used as microwave sources, provides a basic design building block for detection in future atom chip-based compact quantum inertial sensors.