Now here's a topic that gets very little attention and is considered "proven" even moreso than evolution and a billions of years old earth. We have the Protestants to thank for keeping the candle burning on Creationism and for giving it a certain credibility, after Catholics totally abandoned the defence of Creationism over the last 40 years. But since the Protestants have ignored Geocentrism, there's been really no one to defend it, and hence no real discussion.
We freely admit we've not studied this question in any detail at all and therefore are not writing this to contradict the heliocentric model of the solar system. It's the accepted model and until convinced otherwise, we accept it. However, the possibility of Geocentrism does offer fascinating possibilities for the science fiction writer.
Most write-off the question as irrelevant if they are not castigating proponents of Geocentrism as "retards" (this generally unacceptable-in-polite-company words seems to make a resurgence in these debates). But if we try to cut through all that garbage, it seems that there is a theological relevance. We also don't think it's completely cracked, since the observation of motion is always relative (think of how the moon appears to follow you as you drive, or how when on a train the landscape appears to move). Also, in something as massive as the universe, how can anyone say what is, or isn't "the centre" (if we take Geocentrism to mean simply that the earth is centre of the universe, not necessarily that the Ptolemaic model of the solar system is accurate). Until recently the Church seems to have used its teaching authority to hold to Geocentrism -- and no Catholic (and even non-Catholic) can easily ignore the teaching authority of the Church. Not even in science, for the Church has never been some backwards luddite/anti-science institution, but rather quite the opposite (see chapter 5 of How The Catholic Church Built Western Civilization by Dr. Thomas E. Woods, Jr. for a good exposition on this; although we don't endorse the entire work and believe chapter 8 -- on economics -- is totally off-base)
On a more practical level, look up into the clear night sky (you may have to get out of the city to do this) and consider the Earth fixed and unmoving at the centre of creation with the universe in rotation around it. Then, with your eyes still on the sky, imagine we are on a small rock hurtling through space in some backwater galaxy in an infinitely expanding void. You may soon realise why this is no longer an insignificant question.
But as writers, it makes for some interesting ideas. Perhaps given the universal acceptance of heliocentrism one would have to do it in a steampunk or alternate history/space fantasy setting. It seems that all the celestial bodies would have to be a lot closer to earth than we thought, making it a lot easier/faster to get to at least the other planets in the solar system. We don't know enough about Geocentric theory to know whether extrasolar planets are possible under that model. But if they are, then certainly they'd be much closer as well -- making interstellar travel a whole lot cheaper and easier even without faster-than-light technology.