Tunable photonic bandgap (PBG) microstructure fibers, which were filled nematic liquid crystals (NLC), were theoretically investigated based on bandgap theory. By means of the modified plane-wave method, it is found that PBGs shift to the longer wavelength with increasing refractive index of NLC $\left[n_y\left(\theta \right)\right]$ for y-polarized light. Fundamental modes are found in these PBG reigns, whose effective mode area, leakage loss and group velocity dispersion (GVD) have been calculated by using the full-vector finite-element method with anisotropic perfectly matched layers. The mode fields become larger with the increase of $n_y\left(\theta \right)$ , whereas the leakage loss varies slightly. Moreover, GVD is strongly dependent on $n_y\left(\theta \right)$ and wavelength, which is much larger than the material dispersion of silica.