Fourier acceleration is a technique used in Hybrid Monte Carlo simulations to decrease the autocorrelation between subsequent field configurations in the generated ensemble. It has been shown, in the perturbative limit, to eliminate the problem of critical slowing down in a $\phi^4$ theory. As a result, there are several techniques that are being explored to generalize Fourier acceleration to work with non-Abelian gauge theories like QCD. It is hoped that these methods will prove effective at overcoming the problem of critical slowing down, even in the non-perturbative limit. In our work, we show that Fourier acceleration can be applied effectively to a linear sigma model in the symmetry broken phase, leading to reduced autocorrelation and faster thermalization. We present an algorithm for estimating the optimal Fourier acceleration masses dynamically, based on the lattice data. In the future, we hope to explore the effectiveness of these techniques in the strongly-interacting case. Since our $\phi^4$ theory is a linear chiral effective theory for QCD, this could be interesting for those who are seeking to generalize Fourier acceleration to QCD.