Since most current vulnerability analysis methods are only applicable for one-dimensional seismic analysis, we established three-dimensional (3D) seismic damage index functions of bridge risk components. Based on the reliability theory of engineering structure and the method of probability and statistics, a general 3D seismic vulnerability calculation theory of bridge structure was derived and the corresponding analysis method was developed. A typical rigid frame continuous beam combined railway bridge from the Western region was selected as the research object and the vulnerability analysis of the bridge was carried out. Based on the results of the cloud method, the correctness of the 3D seismic vulnerability analysis method was verified and full-view vulnerability cloud pictures of each bridge risk component were established to evaluate the general seismic performance of the bridge structure in 3D earthquakes. Results show that only analysis of the distribution characteristics of the longitudinal and transverse vulnerability of the bridge is not effective for reflecting its actual seismic capacity, so it is necessary to conduct the 3D vulnerability analysis. The full-view vulnerability cloud pictures of the bridge can be applied to evaluate the seismic performance of the bridge, which provides evidence for 3D bridge seismic design.