  We study the zero locus of the Futaki invariant on K-polystable Fano
threefolds, seen as a map from the K\"ahler cone to the dual of the Lie algebra
of the reduced automorphism group. We show that, apart from families 3.9, 3.13,
3.19, 3.20, 4.2, 4.4, 4.7 and 5.3 of the Iskovskikh-Mori-Mukai classification
of Fano threefolds, the Futaki invariant of such manifolds vanishes identically
on their K\"ahler cone. In all cases, when the Picard rank is greater or equal
to two, we exhibit explicit 2-dimensional differentiable families of K\"ahler
classes containing the anti-canonical class and on which the Futaki invariant
is identically zero. As a corollary, we deduce the existence of non
K\"ahler-Einstein cscK metrics on all such Fano threefolds.
