We describe a new sensitive acoustic method to measure nonlinear viscoelastic properties ofrocks directly in the time domain. Unlike static stress-strain measurements, or dynamic acousto-elasticmethods that use steady state resonance to load a strain field into a rock, our method uses twocopropagating longitudinal acoustic waves, one to perturb the strain in the sample and the other to probethe effect of that strain. Experiments on Crab Orchard Sandstone and Lucite samples are presented. Weevaluate the time-dependent nonlinear effects and find strong evidence that rocks have a short-periodnonlinear viscoelastic memory that is a function of the time history of the loaded strain. We develop aphenomenological model to describe both this nonlinear viscoelastic memory and the nonlinear elasticbehavior of rock. Our model shows that the viscoelastic memory is controlled by both the traditionalnonlinear elastic coefficients and a memory strength parameter. These new observations and methods havesignificance for quantifying changes in microstructure. This short-term memory effect, on the order of a small fraction of a cycle, is shown to be related through viscoelasticity to absorption in the sample.