In ancient times, the arguments for geocentrism were primarily from observation. It looks like the heavens revolve around the Earth, so "heavens revolve around Earth" is a reasonable first model. Subsequent theory finds that a better model that more closely matches observations of heavenly bodies is that the Earth is simply rotating. The argument about which celestial body was at the 'center' of the universe then raged for some time. Modern physics has revealed that motion is relative to a reference frame, and that imposing a fixed coordinate system with respect to the relative motions of bodies is actually a poor model.
However, our knowledge of the large-scale static-ish distribution of matter in the universe is largely driven by observation. The observations we have been able to make in modern times of the deep universe suggest that the universe is, on average, evenly distributed around our particular location. Since the universe appears to be, on average, uniform, a reasonable model is that it appears that way from every point in the universe.
Nevertheless, since we are confined to our tiny bubble, it is unlikely that this hypothesis will be tested anytime soon. Relating this back to the original question, it seems that, currently, it looks like the Earth is at the center of the universe. We are unable to verify the hypothesis that the universe would look more-or-less this way from every point experimentally. So, what are some modern arguments against a geocentric (or heliocentric, or, more generally, an "our-local-neighborhood-centric") viewpoint? Specifically, are there any counterpoints, backed up by experiment, to the argument that "it looks like we're at the center, therefore we are at the center"? Alternatively, what specific, tested predictions of prevailing models is this incompatible with?