  The square Bi layers in $A$MnBi$_2$ ($A =$ Sr, Ca) host Dirac fermions which
coexist with antiferromagnetic order on the Mn sublattice below $T_\mathrm{N} =
290\,$K (Sr) and $270\,$K (Ca). We have measured the spin-wave dispersion in
these materials by triple-axis neutron spectroscopy. The spectra show
pronounced spin gaps of 10.2(2)$\,$meV (Sr) and 8.3(8)$\,$meV (Ca) and extend
to a maximum energy transfer of 61 - 63$\,$meV. The observed spectra can be
accurately reproduced by linear spin-wave theory from an Heisenberg effective
spin Hamiltonian. Detailed global fits of the full magnon dispersion are used
to determine the in-plane and inter-layer exchange parameters as well as on the
magnetocrystalline anisotropy constant. To within experimental error we find no
evidence that the magnetic dynamics are influenced by the Dirac fermions.
