We study the existence of strong Kähler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures J on solvmanifolds G/Γ providing some negative results for some classes of solvmanifolds. In particular, we show that if either J is invariant under the action of a nilpotent complement of the nilradical of G or J is abelian or G
Nearby in the stack
is almost abelian (not of type (I)), then the solvmanifold
G/Γ
cannot admit any symplectic form taming the complex structure
J
, unless
G/Γ
is Kähler. As a consequence, we show that the family of non-Kähler complex manifolds constructed by Oeljeklaus and Toma cannot admit any symplectic form taming the complex structure.