A propagator method referred to as third-order algebraic-diagrammatic construction [ADC(3)] for the direct computation of electronic excitation energies and transition moments is presented. This approach is based on a specific reformulation of the diagrammatic perturbation expansion for the polarization propagator, and extends the existing second-order [ADC(2)] scheme to the next level of perturbation theory. The computational scheme combines diagonalization of a Hermitian secular matrix and perturbation theory for the matrix elements. The characteristic properties of the method are compact configuration spaces, regular perturbation expansions, and size-consistent results. The configuration space is spanned by singly and doubly excited states, while the perturbation expansions in the secular matrix extend through third order in the p-h block, second order in the p-h/2p-2h coupling block, and first order in the 2p-2h block. While the simpler ADC(2) method, representing a counterpart to the MP2 (second-order Moller-Plesset) ground-state method, recommends itself for application to larger molecules, the ADC(3) scheme is aimed at a more accurate description of molecular excitation spectra. The relationship of the ADC(3) scheme with coupled cluster methods is discussed, focusing here in particular on the treatment of transition moments.