The microscopic origin of the in-plane (G(x), G(y)) and out-of-plane (G(z)) Dzialoshinsky-Moriya (DM) exchange parameters is considered for the Cu-3(II) clusters. For the systems with the d(x2-y2) ground state of the Cu ions, only Z components of the pair DM exchange parameters are active (G(z) not equal 0, G(x,y) = 0) in the cases of the orientations of the local anisotropy axes z(i) parallel (z(i)parallel to Z) and perpendicular (z(i)perpendicular to Z, x(i)parallel to(-Z)) to the molecular trigonal Z axis. The dependences of the G(x), G(y), and G(z) DM exchange parameters on the tilt of the local magnetic orbitals were obtained for the antiferromagnetic (AFM) clusters with the d(x2-y2) and d(z2) ground state of the Cu ions. The tilt of the local d(x2-y2) orbitals results in the change of the G(z) parameter and appearance of the in-plane DM exchange interactions (G(x) or/and G(y) parameters). The dependence of the G(z) and G(x), G(y) DM exchange parameters on the tilt angle is essentially different. The in-plane DM exchange coupling (G(x), G(y) parameters) can significantly exceed the out-of-plane DM coupling (G(z) parameter). The nonzero G(z) and G(x), G(y) parameters can be positive or negative. For the {Cu-3} nanomagnet with the d(x2-y2) ground state and relatively strong DM coupling, the model explains the three DM exchange parameters of the same value (vertical bar G(z)vertical bar = vertical bar G(x)vertical bar = vertical bar G(y)vertical bar) by the small tilt of the local anisotropy axes z(i) of the CuO4 local groups of the trimer from the positions z(i)perpendicular to Z. The dependence of the DM exchange parameters (G(z), G(x), G(y)) on the tilt for the AFM Cu-3 clusters with the d(z2) ground states of the Cu ions differs significantly from that for the AFM systems with the ground state d(x2-y2) of the individual ions. Large in-plane DM exchange parameters G(x) or/and G(y) result in the mixing of the 2(S = 1/2) and S = 3/2 states and zero-field splitting (ZFS) 2D(DM) of the excited S = 3/2 state. The DM exchange contribution 2D(DM) to ZFS of the excited S = 3/2 state possesses the significant dependence on the tilt of the local magnetic orbitals.