Topological magnons are emergent quantum spin excitations featured by magnon bands crossing linearly at the points dubbed nodes, analogous to fermions in topological electronic systems. Experimental realisation of topological magnons in three dimensions has not been reported so far. Here, by measuring spin excitations (magnons) of a three-dimensional antiferromagnet Cu3TeO6 with inelastic neutron scattering, we provide direct spectroscopic evidence for the coexistence of symmetry-protected Dirac and triply degenerate nodes, the latter involving three-component magnons beyond the Dirac-Weyl framework. Our theoretical calculations show that the observed topological magnon band structure can be well described by the linear-spin-wave theory based on a Hamiltonian dominated by the nearest-neighbour exchange interaction J1. As such, we showcase Cu3TeO6 as an example system where Dirac and triply degenerate magnonic nodal excitations coexist, demonstrate an exotic topological state of matter, and provide a fresh ground to explore the topological properties in quantum materials.