Summary: The Greeks likely rejected a heliocentric theory because it would conflict with the lack of any visible stellar parallax, not for egotistical, common-sense, or aesthetic reasons.

I had always heard that the Greeks embraced a geocentric universe for common-sense, aesthetic reasons - not scientific ones. But it seems as if the real story is more complicated than that:

From Isomorphismes:

Now this is the kicker in your Popperian dirtsack. The Greeks had the right theory (heliocentric solar system) but discarded it on the basis of experimental evidence! Never preach to me about progress-in-science when all you’ve heard is a one-liner about Popper and the communal acceptance of general relativity. Especially don’t follow it up by saying that science marches toward the Truth whilst religion thwarts its progress. According to Astronomer Lisa, it’s not true that the Greeks simply thought they and their Gods were at the centre of the Universe because they were egotistical. They reasoned to the geocentric conclusion based on quantitative evidence. How? They measured parallax.(Difference in stellar appearance from spring to fall, when we’re on opposite sides of the Sun.) Given the insensitivity of their measurement tools at the time, the stars didn’t change positions at all when the Earth moved to the other side of the Sun. Based on that, they rejected the heliocentric hypothesis. If the Earth actually did move around the Sun, then the stars would logically have to appear different from one time to another. But they remain ever fixed in the same place in the Heavens, therefore the Earth must be still (geocentric).

I dug a little bit deeper, and this seems to be more or less accurate. From The Greek Heliocentric Theory and its Abandonment:

This paper then examines possible reasons for the Greek abandonment of the heliocentric theory and concludes that there is no reason to deplore its abandonment. In developing the heliocentric theory the Greeks had run the gamut of theorizing. We are indebted to the Alexandrians and Hipparchus for turning away from speculation to take up the recording of precise astronomical data. Here was laid the foundation upon which modern astronomy was built.

Let  us now suppose that Aristarchus’ theory was widely circulated and that it was given careful consideration by leading astronomers. There is one objection that immediately  arises when the earth is put in motion, the very difficulty which must  have disquieted Copernicus and which caused Tycho Brahe shortly  afterwards to renounce Copernicus’ heliocentric system and to put the earth again at rest. (Tycho reverted to a system first suggested by some ancient Greek, who made the planets revolve about the sun and the sun about the earth.) The difficulty is this. As soon as the  earth is set in motion  in an annual revolution  about the sun, the distance between any two of the earth’s positions that are six months apart will be twice as  great as the earth’s distance from the sun. Over such vast distances some displacement in the positions of the stars ought to be observed. The more accurate the astronomical instruments and the greater the estimated distance of the sun, the more reason should there be to expect stellar displacement. Now it so happened that Aristarchus reached his conclusions at the very time when interest was keen at Alexandria and elsewhere in the Greek world in accurate observations and when marked improvements were being made in precision instruments. To appreciate these developments we need only recall the careful stellar catalogues of Aristyllus and Timocharis early in the third century B.C., the work of the latter enabling Hipparchus to discover the precession of the equinoxes, and the armillary sphere of Eratosthenes by which he was able to  determine  the obliquity of the ecliptic and the circumference of the  earth. Hipparchus continued to make improvements in the next century. He, as we shall  see, had a much better appreciation of the sun’s great distance than Copernicus. Of course it was impossible to observe stellar displacement without the aid of a telescope. Inability to observe it left astronomers with only two alternatives: either the stars were so remote that it was impossible to detect displacement, or the earth would have to remain at rest.

..Heath was of  the opinion that Hipparchus was responsible for the  death of Aristarchus’ theory, that the adherence of so preeminent an astronomer to a geocentric orientation sealed the doom of the heliocentric theory. This is a reasonable conjecture. Hipparchus was  noted for his careful observations, his stellar catalogues, and the remarkable precision of his recordings of solar and  lunar  motions. According to Ptolemy he was devoted to truth above all else and  because he did not possess sufficient data, he refused to attempt to account for planetary motions as he had for those of the sun and moon. His discovery of the precession of the equinoxes attests to the keenness of his observations. He came much closer to appreciating the vast distance of the  sun than Copernicus did.

..We do not know whether or not Hipparchus ever seriously entertained  Aristarchus’ views about the earth’s motions, but from what we have seen of his cautious and accurate methods, it is likely that he would have quickly rejected the heliocentric theory in the absence of visible stellar displacement.

And from The Ancient Greek Astronomers: A Remarkable Record of Ingenuity:

Aristarchus was successful in explaining variations in brilliance and reverse courses of the planets, but planetary motions are far more complicated than that. Kepler was the first to realize that the planets do not describe circular orbits, but rather ellipses, and that the sun is not in the middle of these orbits but in the foci of the ellipses. That something was wrong might have been suspected as early as 330  B.C., for Callippus noticed that the seasons were not of the same length. He estimated their lengths between solstices and equinoxes to  be 94, 92, 89, and 90 days- figures that are very nearly correct. Or to show the irregularities that might result from combining the eccentricities of the orbits of two  planets, in  some years Mars and the earth at closest approximation are 36 million miles apart and in other years (as in 1948) may be 63 million miles apart at their nearest approach. Now the Alexandrians were pointing their precision sights at the planets and must have been disturbed by these peculiarities. Furthermore they would have been less kindly disposed towards Aristarchus’ explanation of the absence of visible stellar parallax by placing the stars at  an almost infinite distance away because they had a better appreciation of the sun’s vast distance and consequently would have stronger reason to expect to find parallax. It would seem that the more precise the  instruments, the  less  likelihood there would be of the earth’s being in motion.