Starting from Housner's seminal work on the rocking motion of a single rigid block, the dynamic behavior of masonry structures has been broadly investigated, as instrumental for a reliable seismic vulnerability assessment. Among computational approaches available in the literature, the attention is focused here on the Non-Smooth Contact Dynamics (NSCD) method, originally proposed to predict the dynamic behavior of general collections of rigid bodies in contact with each other under external loadings. Attractive of the method is its capability to avoid the need for event detection typical of event-driven approaches. That is obtained by addressing a time-integrated version of the equations of motion and interpreting the hard-contact law in a time-averaged sense. Several applications of the method have proven its merit when considering historical masonry block structures, i.e. characterized by unilateral and dry-frictional contacts. Following previous contributions dealing with 2D problems, a variational-based NSCD approach is proposed for 3D masonry block structures subject to seismic excitation, which achieves computational efficiency thanks to the solution, at each time step, of a second-order conic programming problem. Numerical results are discussed to test the performances of the approach.