To measure the frequency dependence of complex permittivity of a dielectric rod inserted in the center of a cylindrical cavity, a new method using the multi-mode TM0m0 of the cavity is developed on the basis of the rigorous analysis, where the effect of sample insertion holes is taken into account. Three cavities using the TM010 and TM020 modes and two cavities using the TM010, TM020, TM030 and TM040 modes are designed and manufactured with different sizes. These five cavities enable frequency dependence measurements of complex permittivity of a fused silica rod sample over eight octave bands from 3 to 26 GHz. The amount of scatter in the measured values of the relative permittivity and loss tangent are within 0.5% and 10%, respectively.