Magnons in ferromagnets have one chirality, and typically are in the GHz range and have a quadratic dispersion near the zero wave vector. In contrast, magnons in antiferromagnets are commonly considered to have bands with both chiralities that are degenerate across the entire Brillouin zone, and to be in the THz range and to have a linear dispersion near the center of the Brillouin zone. Here we theoretically demonstrate a new class of magnons on a prototypical $d$-wave altermagnet ${\mathrm{RuO}}_{2}$ with the compensated antiparallel magnetic order in the ground state. Based on density-functional-theory calculations we observe that the THz-range magnon bands in ${\mathrm{RuO}}_{2}$ have an alternating chirality splitting, similar to the alternating spin splitting of the electronic bands, and a linear magnon dispersion near the zero wave vector. We also show that, overall, the Landau damping of this metallic altermagnet is suppressed due to the spin-split electronic structure, as compared to an artificial antiferromagnetic phase of the same ${\mathrm{RuO}}_{2}$ crystal with spin-degenerate electronic bands and chirality-degenerate magnon bands.