This paper presents a new design method of a planar 3 degrees-of-freedom(DOF) serial linkage-type vibration energy harvester with a single proof mass. The harvester is designed to generate electrical power at equally spaced three target resonant frequencies which can be chosen arbitrarily. For given target frequencies and a proof mass, the design method involves (1) the determination of the stiffness matrix, (2) the synthesis of the stiffness by means of a parallel connection of three line springs and (3) its conversion into a 3DOF device connected serially by torsional springs. The torsional springs are realized by the flexible hinge joints and the polyvinylidene fluoride(PVDF) films are attached on the joints. Upon determination of the desired stiffness matrix, the SQP algorithm is utilized to find the optimum locations and spring constants of the serial hinge joints for the minimum difference among three electrical power peaks. The FEM analysis and experiments are conducted to verify the proposed design method. Three measured resonant power peaks occur at 24.7, 30.4 and 33.6 Hz comparing to the target frequencies of 25, 30 and 35 Hz. The normalized maximum power of $14.5\,{\rm{uW}}/{({{\rm{ms}}}^{-2})}^{2}$ is generated at 24.7 Hz. The experimental results also demonstrate that the harvester can generate at least 18.6% of the peak power throughout the frequency range from 23.1 to 36.5 Hz, which ensures consistently acquirable power within the operating frequency range by virtue of the coupled effect of a serial linkage-type structure.