The principle of quantitative acoustic microscopy (QAM) is to form two-dimensional acoustic parameter maps from a collection of radiofrequency (RF) signals acquired by raster scanning a biological sample. Despite their relatively simple structure, i.e. two main reflections, QAM RF signals are currently sampled at very high frequencies, e.g., at 2.5 GHz for QAM system employing a single-element transducer with a center frequency of 250-MHz. The use of such high sampling frequencies is challenging because of the potentially large amount of acquired data and the cost of the necessary analog to digital converters. In this work, we propose a sampling scheme based on the finite rate of innovation theory that exploits the limited numbers of degrees of freedom of QAM RF signals and allows the reconstruction of accurate acoustic maps from a very limited number of samples.