The special kind of diffraction radiation from the uniformly moving electron beam in a periodical waveguide structure has great potential applications in developing the tunable terahertz radiation sources. Rigorous theoretical analyses and detailed computer simulation on the diffraction radiation from the periodical waveguide structure excited by the eccentric electron beam are carried out. Our results show that the eccentric beam can primarily excite the axial symmetrical modes (TM0 modes) and axial unsymmetrical modes with axial variation numbers of 1 and 2. The energies of the unsymmetrical modes increase with the distance of electron beam to the axis. For each mode, the radiation intensity changes with beam location as the square of modified Bessel function and in direct proportion with the square of the charge quantity. The results of theoretical analyses and computer simulations are in good agreement with each other. These results are of significance for developing this kind of radiation source.