The germanium-on-insulator (GOI) has recently emerged as a new platform for complementary metal-oxide-semiconductor (CMOS)-compatible photonic integrated circuits. Here we report on resonant-cavity-enhanced optical responses in Ge photodetectors on a GOI platform where conventional photodetection is difficult. A 0.16% tensile strain is introduced to the high-quality Ge active layer to extend the photodetection range to cover the entire range of telecommunication C- and L-bands (1530-1620 nm). A carefully designed vertical cavity is created utilizing the insulator layer and the deposited SiO2 layer to enhance the optical confinement and thus optical response near the direct-gap absorption edge. Experimental results show a responsivity peak at 1590 nm, confirming the resonant cavity effect. Theoretical analysis shows that the optical responsivity in the C- and L-bands is significantly enhanced. Thus, we have demonstrated a new type of Ge photodetector on a GOI platform for CMOS-compatible photonic integrated circuits for telecommunication applications.