Ultrasound elastography infers mechanical properties of living tissues from ultrasound radiofrequency (RF) data recorded while the tissues are undergoing deformation. A challenging yet critical step in ultrasound elastography is to estimate the tissue displacement (or, equivalently the time delay estimate) fields from pairs of RF data. The RF data are often corrupted with noise, which causes the displacement estimator to fail in many in vivo experiments. To address this problem, we present a nonlocal, coherent denoising approach based on Bayesian estimation to reduce the impact of noise. Despite incoherent denoising algorithms that smooth the B-mode images, the proposed denoising algorithm is used to suppress noise while maintaining useful information such as speckle patterns. We refer to the proposed approach as COherent Denoising for Elastography (CODE) and evaluate its performance when CODE is used in conjunction with the two state-of-art elastography algorithms, namely: (i) GLobal Ultrasound Elastography (GLUE) and (ii) Dynamic Programming Analytic Minimization elastography (DPAM). Our results show that CODE substantially improves the strain result of both GLUE and DPAM.