  We prove that for a compact K\"ahler threefold with canonical singularities
and vanishing first Chern class, the projective fibres are dense in the
semiuniversal deformation space. This implies that every K\"ahler threefold of
Kodaira dimension zero admits small projective deformations after a suitable
bimeromorphic modification. As a corollary, we see that the fundamental group
of any K\"ahler threefold is a quotient of an extension of fundamental groups
of projective manifolds, up to subgroups of finite index.
  In the course of the proof, we show that for a canonical threefold with $c_1
= 0$, the Albanese map decomposes as a product after a finite \'etale base
change. This generalizes a result of Kawamata, valid in all dimensions, to the
K\"ahler case. Furthermore we generalize a Hodge-theoretic criterion for
algebraic approximability, due to Green and Voisin, to quotients of a manifold
by a finite group.
