  We study the topological properties of magnon excitations in
three-dimensional antiferromagnets, where the ground state configuration is
invariant under time-reversal followed by space-inversion ($PT$-symmetry). We
prove that Dirac points and nodal lines, the former being the limiting case of
the latter, are the generic forms of symmetry-protected band crossings between
magnon branches. As a concrete example, we study a Heisenberg spin model for a
"spin-web" compound, Cu$_3$TeO$_6$, and show the presence of the magnon Dirac
points assuming a collinear magnetic structure. Upon turning on
symmetry-allowed Dzyaloshinsky-Moriya interactions, which introduce a small
non-collinearity in the ground state configuration, we find that the Dirac
points expand into nodal lines with nontrivial $Z_2$-topological charge, a new
type of nodal lines unpredicted in any materials so far.
