We introduce a new finite-difference time-domain (FDTD) method for solving the TM (transverse magnetic) mode reduction of Maxwell's equations with complicated material interfaces. The method uses an unstructured quadrilateral mesh intended to conform to the complicated geometry of material interfaces. Using a quadrilateral to unit-square local coordinate transformation, we remap all the field components from the physical grid to a computational grid consisting of unit-squares. On the computational grid of unit-squares we update the fields in time using the usual FDTD field updates. The local coordinate remap, however, changes a scalar material on the physical space to an anisotropic material on computational space; we handle the mixed components of the anisotropic material by interpolating the non-colocated field components. Finally, we resolve material parameter jumps across interfaces using a harmonic-mean. By testing our method on the TM0,1 mode in partially filled infinite cylindrical cavity, we verify that the method converges, is second order accurate, and maintains stability.