Given a polynomial dynamical system on real affine space, we show that the forward orbit of any point intersects any algebraic subvariety in at most finitely many infinite arithmetic progressions, with the remaining points of intersection lying in a set of (Banach) density zero. This may be viewed as a weak asymptotic version of the Dynamical Mordell-Lang Conjecture. The result actually holds for dynamical systems on any affine variety over any field in arbitrary characteristic. The proof uses methods of ergodic theory applied to compact Berkovich spaces, in particular a strong version of the Poincare recurrence theorem due to Furstenberg.