Spin-dynamics techniques have been used to perform large-scale simulations of the dynamic behavior of the classical Heisenberg antiferromagnet in simple cubic lattices with linear sizes $L<~60.$ This system is widely recognized as an appropriate model for the magnetic properties of ${\mathrm{RbMnF}}_{3}.$ Time evolutions of spin configurations were determined numerically from coupled equations of motion for individual spins using an algorithm implemented by Krech et al., which is based on fourth-order Suzuki-Trotter decompositions of exponential operators. The dynamic structure factor was calculated from the space- and time-displaced spin-spin correlation function. The crossover from hydrodynamic to critical behavior of the dispersion curve and spin-wave half-width was studied as the temperature was increased towards the critical temperature. The dynamic critical exponent was estimated to be $z=(1.43\ifmmode\pm\else\textpm\fi{}0.03),$ which is slightly lower than the dynamic scaling prediction, but in good agreement with a recent experimental value. Direct, quantitative comparisons of both the dispersion curve and the line shapes obtained from our simulations with very recent experimental results for ${\mathrm{RbMnF}}_{3}$ are presented.