Closed-room scenarios are characterized by reverberation, which decreases the performance of applications such as hands-free teleconferencing and multichannel sound reproduction. However, exact knowledge of the sound field inside a volume of interest enables the compensation of room effects and allows for a performance improvement within a wide range of applications. The sampling of sound fields involves the measurement of spatially dependent room impulse responses, where the Nyquist-Shannon sampling theorem applies in the temporal and spatial domains. The spatial measurement often requires a huge number of sampling points and entails other difficulties, such as the need for exact calibration of a large number of microphones. In this paper, a method for measuring sound fields using moving microphones is presented. The number of microphones is customizable, allowing for a tradeoff between hardware effort and measurement time. The goal is to reconstruct room impulse responses on a regular grid from data acquired with microphones between grid positions, in general. For this, the sound field at equidistant positions is related to the measurements taken along the microphone trajectories via spatial interpolation. The benefits of using perfect sequences for excitation, a multigrid recovery, and the prospects for reconstruction by compressed sensing are presented.