Bouncing cosmologies are often proposed as alternatives to standard inflation for the explanation of the homogeneity and flatness of the universe. In such scenarios, the present cosmological expansion is preceded by a contraction phase. However, during the contraction, in general the anisotropy of the universe grows and eventually leads to a chaotic mixmaster behavior. This would either be hard to reconcile with observations or even lead to a singularity instead of the bounce. In order to preserve a smooth and isotropic bounce, the source for the contraction must have a super-stiff equation of state with P/ρ = w > 1. In this letter we propose a new mechanism to solve the anisotropy problem for any low-energy value of w by arguing that high energy physics leads to a modification of the equation of state, with the introduction of non-linear terms. In such a scenario, the anisotropy is strongly suppressed during the high energy phase, allowing for a graceful isotropic bounce, even when the low-energy value of w is smaller than unity.