The transverse piezoelectric coefficient d(31), has been calculated for the six domain-engineered structures occurring in perovskite single crystals, using data for rhombohedral PMN-33PT [0.67Pb(Mg1/3Nb2/3)O-3-0.33PbTiO(3)], orthorhombic potassium niobate (KNbO3), tetragonal barium titanate (BaTiO3), and tetragonal lead titanate (PbTiO3). Unlike the longitudinal coefficient (d(33)), d(31)* is found to be strongly dependent on the transverse (x(1)') direction of the as-cut crystal. In general, different domains in a domain-engineered structure will contribute different values of d(31)* to that measured. Predicting the global d(31), is therefore difficult since it will depend on the proportion of each domain variant in the structure. Important qualitative differences between tetragonal BaTiO3 and PbTiO3 are discussed. Whereas polarization rotation is important in BaTiO3, PbTiO3 shows a stronger collinear piezoelectric effect due the absence of a low-temperature ferroelectric-ferroelectric phase transition. This leads to low values of d(33) and even positive values of d(31)*, in the 111-poled (C: pseudocubic) domain-engineered structure. The methodology described can be usefully applied to all perovskites. (c) 2005 American Institute of Physics.