The best effective estimate I know of in general is the very recent and impressive work of [Einsiedler-Venkatesh-Margulis.][1]   They provide polynomial decay in the ergodic theorem (for unipotent and other flows) on a set of large (actually estimated) measure for quotients of most lie groups by arithmetic lattices.

There were previously things like this: for hyperbolic surfaces, on a set of large (controlled) measure, the error for the time average of the horocycle flow of length $T$ is something like $S(f) T^{-\epsilon}$ where $f$ is the smooth function you're averaging, $S(f)$ is a Sobolev norm, and $\epsilon$ depends on the spectrum of the surface.

See the nice [survey (pdf)][2] by Einsiedler.


I would be very interested in any better estimates for flows on arithmetic hyperbolic $3$-manifolds given information on the trace field, quaternion algebra, et cetera.


  [1]: http://arxiv.org/abs/0708.4040
  [2]: http://www.math.ohio-state.edu/~manfred/eff-survey.pdf