Scanning acoustic microscopy is a well-accepted modality for forming quantitative 2D maps of acoustic properties of soft tissues at microscopic scales. In our studies, the sample is raster-scanned with a spatial step size of 2 μm using a 250 MHz transducer resulting in 3D RF data cubes. Each RF signal is processed to obtain, for each spatial location, acoustic parameters, e.g., the speed of sound. The scanning time is directly dependent on the sample size and can range from few minutes to hours. In order to maintain constant experimental conditions for the sensitive thin sectioned samples, the scanning time is an important practical issue. Hence, the main objective of this work is to reduce the scanning time by reconstructing acoustic microscopy images from spatially under sampled measurements, based on the theory of compressive sampling. A recently proposed approximate message passing method using a Cauchy maximum a posteriori image denoising algorithm is thus employed to account for the non-Gaussianity of quantitative acoustic microscopy wavelet coefficients.