A novel framework for compressive sensing (CS) data acquisition and reconstruction in quantitative acoustic microscopy (QAM) is presented. Three different CS patterns, adapted to the specifics of QAM systems, were investigated as an alternative to the current raster-scanning approach. They consist of diagonal sampling, a row random, and a spiral scanning pattern and can all significantly reduce both the acquisition time and the amount of sampled data. For subsequent image reconstruction, we design and implement an innovative technique, whereby a recently proposed approximate message passing method is adapted to account for the underlying data statistics. A Cauchy maximum a posteriori image denoising algorithm is thus employed to account for the non-Gaussianity of QAM wavelet coefficients. The proposed methods were tested retrospectively on experimental data acquired with a 250- or 500-MHz QAM system. The experimental data were obtained from a human lymph node sample (250 MHz) and human cornea (500 MHz). Reconstruction results showed that the best performance is obtained using a spiral sensing pattern combined with the Cauchy denoiser in the wavelet domain. The spiral sensing matrix reduced the number of spatial samples by a factor of 2 and led to an excellent peak signal-to-noise ratio of 43.21 dB when reconstructing QAM speed-of-sound images of a human lymph node. These results demonstrate that the CS approach could significantly improve scanning time, while reducing costs of future QAM systems.