Modern arguments against Geocentrism? 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?
 A: 
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?

There are a vast number of statements that one could make that one cannot reasonably rule out observationally. We can only see so far, and anything could be behind our cosmological horizon. It's not impossible that the Earth sits exactly at the center of a structure that is far larger than our Hubble sphere, and we would be none the wiser. It's just another Russell's teapot.
As a practical, working assumption, we take the entire universe to be described by the FLRW metric, which has no "center" for the Earth to be at. This is a mathematically simple option that fits the data. However, the crucial thing is that, even though the FLRW metric extrapolates beyond our Hubble patch, it is only used to make statements about our Hubble patch. In other words, the validity of conventional cosmology only require our universe to be locally like the FLRW metric, not globally, and we can test the former assumption perfectly well. (It is true that people sometimes elide the two in pop science, but just about everybody in the field knows the distinction, at least tacitly.)
The point is, cosmology doesn't make definite statements about your argument. But within the current formalism, having the Earth be at the center of a giant unobservable structure would be mathematically clunky, artificial, and unnecessary to explain any current experimental results. 

Incidentally, we could conclude that the Earth was (1) at the center of the universe, (2) not at the center, or that (3) the question was meaningless, if the universe was finite in extent, and small enough to see from one end to the other. We could detect this by seeing multiple images of the same object in different directions in the sky. Depending on the particular shape of the universe and the data, this could support any of the three options. (The actual technique is a bit more complicated, since we can't make out specific objects in the CMB, just cold and warm spots; instead we look for "matched circles" with similar temperature profiles.)
There have been several searches for this kind of structure, with one popular candidate for the shape of the universe being the Poincare sphere. However, to date all such searches have come up negative (for a recent review, see here), which means that either the universe is infinite in extent or finite, but too large to see the shape of. In either case, for now we can't give a definite answer to your question. 
A: 
it seems that, currently, it looks like the Earth is at the center of the universe.

This doesn't really make sense. You haven't defined what it would look like for an observer not to find themselves at the center. E.g., would they observe that light from their own galaxy was Doppler shifted? That would be silly.

We are unable to verify the hypothesis that the universe would look more-or-less this way from every point experimentally.

Not true. We do surveys of galaxies, and we find that on large scales, the universe is homogeneous. This is a falsifiable hypothesis. If we did a more careful survey and detected some inhomogeneity, it would be falsified.
