The crucial task of the diagnosis of an existing masonry structure is to assess the current values of the mechanical parameters of the materials from which the structure was erected—usually bricks and mortar. The article presents the results of minor-destructive tests carried out on bed joints of three-brick-masonry prisms prepared in the laboratory. Three types of mortars used in the masonry were tested, which differ by the type and amount of binder. In order to determine mortar compression strength, three modern diagnostic methods were used: double punch test (DPT), standard penetrometric test (PT) and torque penetrometric test (TPT). Tests were carried out after 4, 12 and 90 weeks. The mortar strength determined in each of these tests was compared with the mortar reference strength determined on the beam specimen according to the methodology given in EN 1015-11. The results of the conducted tests confirmed the high usefulness of all three diagnostic methods. However, limitations in the application of the PT test were noticed—only lime mortars and weak cement–lime mortars can be tested with this method. In the case of mortars with an increased amount of cement binder, the impact energy is too low to estimate the compressive strength of the mortar in the brick wall joint. Technical limitations in the use of TPT and DPT tests were also indicated—weak lime mortars with low cohesion do not allow for obtaining reliable results. It was shown that DPT results strongly depend on two factors, specimen slenderness and mortar strength. Due to this fact, simple non-parameter conversion from mortar compressive strength according to the DPT test into mortar reference strength may lead to significant overestimation. As the results show, in newly built masonry, proper selection of diagnostic method is crucial due to the strong dependence of mortar curing dynamics on its location in the joint. This paper helps to match diagnostic techniques with the condition and type of mortar in the existing structure.