The heart is a dynamic pump whose function is influenced by its mechanical properties. The viscoelastic properties of the heart, i.e. its ability to exhibit both elastic and viscous characteristics upon deformation, influence cardiac function. Viscoelastic properties change during heart failure (HF), but, direct measurements of failing and non-failing myocardial tissue stress relaxation under constant displacement are lacking. Further, how consequences of tissue remodeling, such as fibrosis and fat accumulation, alter the stress relaxation remains unknown. To address this gap, we conducted stress relaxation tests on porcine myocardial tissue to establish baseline properties of cardiac tissue. We found porcine myocardial tissue to be fast relaxing, characterized by stress relaxation tests on both a rheometer and microindenter. We then measured human left ventricle (LV) epicardium and endocardium human tissue from non-failing, ischemic HF, and non-ischemic HF patients by microindentation. We found that the ischemic HF had slower stress relaxation than non-failing endocardium; and that slower stress relaxing tissues were correlated with increased collagen deposition and increased α-smooth muscle actin (α-SMA) stress fibers, a marker of fibrosis and cardiac fibroblast activation, respectively. In the epicardium, we found that ischemic HF had faster stress relaxation than non-ischemic HF and non-failing; and that faster stress relaxation correlated with Oil Red O staining, a marker for adipose tissue. These data show that changes in stress relaxation vary across the different layers of the heart during ischemic vs. non-ischemic HF. These findings reveal how the viscoelasticity of the heart changes, which will lead to better modeling of cardiac mechanics for in vitro and in silico HF models.