The boundary integral method is a powerful method for modeling soft objects such as capsules, vesicles or red blood cells in flows at low Reynolds numbers. The beauty of this method lies in its effective two- dimensional description of the boundaries making fluid-structure inter- action simple in comparison to other methods. So far, the boundary integral method is restricted to cells with a viscoelastic membrane and a purely viscous interior, whereas most biological cells contain a com- plex cytoskeleton and a nucleus in addition to their membrane. Here, we develop an extended three-dimensional boundary integral method, that can handle such cells, which are treated as a homogeneous vis- coelastic material surrounded by a viscoelastic membrane. We use our method to study systematically the influence of bulk and membrane viscoelasticity on the dynamics of a cell in shear flow closely corre- sponding to recent experiments [1]. [1] R. Gerum et al., Viscoelastic properties of suspended cells measured with shear flow deformation cytometry, elife 11, e78823, 2022.