Limit analysis provides a simple yet effective means of verifying the safety of masonry constructions. Of the numerical rigid block-based approaches that have been developed for this purpose, it has been common to model rocking, sliding and/or crushing of the masonry by use of suitable joint interface failure envelopes. However, in this case failure can only occur at joints (or at a limited number of other pre-defined failure planes). Also, as it will seldom be feasible to model every masonry unit in a real physical construction with a rigid block, engineering judgement must usually be exercised on the number of rigid blocks to be employed; if too few blocks are used there is a danger of over-estimating the safety of a given construction. In this contribution discontinuity layout optimization (DLO) is used as a powerful alternative to the traditional rigid block limit analysis method. DLO involves discretizing a solid body using nodes, and then interconnecting node pairs with potential discontinuities. Parametric surfaces are used to describe the geometry, allowing discontinuities to be identified even within curved structures. A homogenized masonry failure envelope is used to enable a masonry construction comprising numerous physical units to be modelled in a computationally efficient manner. Solutions are obtained using optimization, with the critical subset of discontinuities at failure that define the geometry of the critical failure mechanism sought. In the contribution the proposed approach is applied to simple 3D example problems, demonstrating the efficacy and computational efficiency of the proposed DLO-based analysis approach.