We have designed and implemented an approach for three-dimensional (3D) structured illumination (SI) microscopy (SIM) based on a quasi-monochromatic extended source illuminating a Wollaston prism to improve robustness, light efficiency and flexibility over our previous design. We show through analytical and experimental verification of the presented theoretical framework for our proposed tunable structured illumination microscopy (TSIM) system, that a simple and accurate determination of the axial modulation of the SI pattern is achieved, enabling a realistic characterization of the system's effective optical transfer function (OTF). System performance as a function of the extended source size is investigated with simulations. Results from a comparative performance analysis of the proposed TSIM system and traditional SIM systems show some advantages over the traditional two-wave and three-wave interference SIM systems. We show that by controlling the source size and thereby the axial modulation of the 3D SI pattern, the TSIM scheme offers increased OTF compact support and improved optical sectioning capability, quantified by the integrated intensity, under certain conditions, which may be desirable when imaging optically thick samples. The additional tunability of the 3D SI pattern, provides a unique opportunity for OTF engineering in our TSIM system.