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$_3$Snslash 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.
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