X-rays can penetrate deeply into biological cells and thus allow for examination of their internal structures with high spatial resolution. In this study, X-ray phase-contrast imaging and tomography is combined with an X-ray-compatible optical stretcher and microfluidic sample delivery. Using this setup, individual cells can be kept in suspension while they are examined with the X-ray beam at asynchrotron. From the recorded holograms, 2D phase shift images that are proportional to the projected local electron density of the investigated cell can be calculated. From the tomographic reconstruction of multiple such projections the 3D electron density can be obtained. The cells can thus be studied in a hydrated or even living state, thus avoiding artifacts from freezing, drying or embedding, and can in principle also be subjected to different sample environments or mechanical strains. This combination of techniques is applied to living as well as fixed and stained NIH3T3 mouse fibroblasts and the effect of the beam energy on the phase shifts is investigated. Furthermore, a 3D algebraic reconstruction scheme and a dedicated mathematical description is used to follow the motion of the trapped cells in the optical stretcher for multiple rotations.