  We study periodically driven insulating noncollinear stacked kagome
antiferromagnets with a conventional symmetry-protected three-dimensional (3D)
in-plane $120^\circ$ spin structure, with either positive or negative vector
chirality. We show that the symmetry protection of the in-plane $120^\circ$
spin structure can be broken in the presence of an off-resonant circularly or
linearly polarized electric field propagating parallel to the in-plane
$120^\circ$ spin structure (say along the $x$ direction). Consequently,
topological Floquet Weyl magnon nodes with opposite chirality are photoinduced
along the $k_x$ momentum direction. They manifest as the monopoles of the
photoinduced Berry curvature. We also show that the system exhibits a
photoinduced magnon thermal Hall effect for circularly polarized electric
field. Furthermore, we show that the photoinduced chiral spin structure is a
canted 3D in-plane $120^\circ$ spin structure, which was recently observed in
the equilibrium noncollinear antiferromagnetic Weyl semimetals Mn$_3$Sn\slash
Ge. Our result not only paves the way towards the experimental realization of
Weyl magnons and photoinduced thermal Hall effects, but also provides a
powerful mechanism for manipulating the intrinsic properties of 3D topological
antiferromagnets.
