Transmitted-light coherence-controlled holographic microscope (CCHM) based on an off-axis achromatic interferometer allows us to use light sources of arbitrary degree of temporal and spatial coherence. Besides the conventional DHM modes such as quantitative phase contrast imaging and numerical 3D holographic reconstruction it provides high quality (speckle-free) imaging, improved lateral resolution and optical sectioning by coherence gating. Optical setup parameters and their limits for a technical realization are derived and described in detail. To demonstrate the optical sectioning property of the microscope a model sample uncovered and then covered with a diffuser was observed using a low-coherence light source.