We present a new approach for estimation of transmural distributions of stress and strain in the equatorial region of a passive left ventricle. We employ a thick-walled cylindrical geometry, assume that myocardium is incompressible, and use a three-dimensional constitutive relation that yields a material symmetry consistent with observed transmural variations in muscle fiber orientations. Moreover, we consider finite deformations including inflation, extension, twist, and transmural shearing and suggest a new method for determination of the requisite deformation parameters directly from experimental strain data. We show representative transmural distributions of stress and strain, and perform a parametric study to illustrate differing predictions of stress induced by varying boundary conditions, muscle fiber orientations, or modes of deformation. Our analysis can be used to guide and check future predictions of cardiac stresses, and to guide experimentalists by suggesting the accuracy of measurements essential for stress analysis in the heart.