Monte Carlo and spin dynamics techniques have been used to perform large-scale simulations of the dynamic behavior of a nanoscale, classical, Heisenberg antiferromagnet on a simple-cubic lattice with linear sizes $L\ensuremath{\leqslant}40$ at a temperature below the N\'eel temperature. Nanoparticles are modeled with completely free boundary conditions, i.e., six free surfaces, and nanofilms are modeled with two free surfaces in the spatial $z$ direction and periodic boundaries parallel to the surfaces in the $xy$ direction, which are compared to the ``infinite'' system with periodic boundary conditions. The temporal evolutions of spin configurations were determined numerically from coupled equations of motion for individual spins using a fast spin dynamics algorithm with the fourth-order Suzuki-Trotter decomposition of exponential operators, with initial spin configurations generated by Monte Carlo simulations. The local dynamic structure factor $S(q,\ensuremath{\omega})$ was calculated from the local space- and time-displaced spin-spin correlation function. Multiple excitation peaks for wave vectors within the first Brillouin zone appear in the spin-wave spectra of the transverse component of dynamic structure factor ${S}^{T}(q,\ensuremath{\omega})$ in the nanoscale classical Heisenberg antiferromagnet, which are lacking if periodic boundary conditions are used. With the assumption of $q$-space spin-wave reflections with broken momentum conservation due to free-surface confinements, we successfully explained those spectra quantitatively in the linear dispersion region. Meanwhile, we also observed two unexpected quantized spin-wave excitation modes in the spatial $z$ direction in nanofilms for ${S}^{T}(q,\ensuremath{\omega})$ not expected in bulk systems. The results of this study indicate the presence of unexpected forms of spin-wave excitation behavior that have yet to be observed experimentally but could be directly tested through neutron scattering experiments on nanoscale ${\mathrm{RbMnF}}_{3}$ particles or films.