This paper introduces a method to recover unmeasured room impulse responses (RIRs) in acoustical spaces with unknown properties and contents, by means of a compressed sensing methodology. Methods published in the existing literature have been validated in empty, convex rooms; a limited subset of the many, diverse acoustical spaces one can encounter. It results a challenge to represent such diverse wave phenomena with a sparse set of plane waves or equivalent sources, given the coupling between the sparsity of such representations and hypotheses regarding the properties of the acoustical space and its contents, far-field measurement distances, and other parameters. In contrast to this philosophy, the method introduced in this paper exploits the sparsity inherent to the mathematical structure of the wavefronts present in the RIRs, which without further hypotheses carry themselves all the information about the wave propagation in the room. In essence, the measured RIRs are instead represented with a sparse set of curved elementary functions of various sizes, propagation directions and times of arrival, which are linked with the various shapes and locations of the unknown scatterers and boundaries in the room. The main contribution of this work is thus to enable the measurement of RIRs in more complex acoustical spaces, while keeping the number of microphones to a minimum with the use of compressed sensing. The method is formulated as a sparse optimization problem, and the solution is obtained with an iterative thresholding algorithm whose threshold value is determined from the measurements. An analysis of sensing coherence is included, and the performance of the method is experimentally evaluated with 1D microphone array measurements in two lecture rooms and one meet-ing room. For the sake of comparison, the RIRs are also linearly interpolated using a low-pass filter in the wavenumber-frequency domain. The experimental results demonstrate that the proposed method is superior than linear interpolation in all the cases investigated, motivating further development of the method to higher spatial dimensions. In terms of accuracy, the proposed method attains recovery errors in the same order of magnitude as those attained by methods in the literature, yet here the acoustical spaces have arbitrary contents and exhibit more complex geometries and boundary conditions.