The choice of a sustainable construction material needs to take into account not just the environmental impact of the material, but according to the 2030 Agenda for Sustainable Development by the UN, one also needs to consider ease of access, the utilization of locally available materials, and the durability and reliability of the construction itself. Mortared masonry has been used around the world for several hundred years as an accessible type of construction. In masonry mortars, lime and cement are often integrated together for combined advantages: enhanced workability, breathability, and better environmental performance due to the former, and higher strength and shorter setting duration due to the latter. However, despite being extensively studied for their effects on the mechanical properties of mortar, not much is known about the impact of varying lime and cement ratios in the binder on the mechanical performance of masonry as a whole. Variations in the properties of mortars do not always have a significant impact on the mechanical behavior of masonry structures. Therefore, this article details an experimental campaign to measure the compressive strength, E-modulus, flexural strength, and shear bond strength of masonry samples containing two distinct lime–cement mortars (1:2:9 and 1:1:6 cement:lime:sand) and one cement mortar (1:0:5). The results show that more than the presence of lime in the mortar, the strength of the mortar influenced the flexural strength of the masonry ranging from 0.1 to 1.2 MPa. No discernable correlation was observed between the presence of lime in the mortar and the cohesion in the masonry (0.29 to 0.41 MPa). The values of the compressive strength (6.0 to 7.2 MPa) and E-modulus (3.8 to 4.5 GPa) of the masonry decreased and pre-peak ductility increased with an increase in the quantity of lime in the mortar. The recommendations of Eurocode 6 for the flexural strength of the initial shear bond strength were found to be conservative for different mortar strength classes, and significantly unconservative for compressive strength (by 50% to 70%).