  We study the mobility of solitons in second-harmonic-generating lattices.
Contrary to what is known for their cubic counterparts, discrete quadratic
solitons are mobile not only in the one-dimensional (1D) setting, but also in
two dimensions (2D). We identify parametric regions where an initial kick
applied to a soliton leads to three possible outcomes, namely, staying put,
persistent motion, or destruction. For the 2D lattice, it is found that, for
the solitary waves, the direction along which they can sustain the largest kick
and can attain the largest speed is the diagonal. Basic dynamical properties of
the discrete solitons are also discussed in the context of an analytical
approximation, in terms of an effective Peierls-Nabarro potential in the
lattice setting.
