Why did the ancients fail to discover that the Earth orbits the Sun? The ancients observed that the Sun and the 'fixed' stars rotated about the Earth. They were also aware that the Earth was spherical. They performed many astronomical measurements on the planets - which are far less 'obvious' than the Sun and the stars. Presumably they knew that the speed of rotation of the Sun and the 'fixed' stars differed and that difference happened to be one day a year. It would seem (in retrospect) obvious that this was no coincidence and no other heavenly motion was so clearly linked to any other. Could they not have deduced that the difference was due to the annual path of the Earth around the Sun?
 A: Actually, heliocentric models were proposed in ancient times by many different people.  The first of which was Aristarchus of Samos in about 270 BC.  See the Wikipedia article on Heliocentrism for full details.  
One major reason geocentrism presisted, at least up through the time of Copernicus was due to the influence of the Catholic Chruch.  They strongly held that the Earth was the center of creation and at some level attempted to supress opposing ideas.
Another reason that geocentrism held on so long is that we can't physically feel the Earth moving.  To our natural senses, the Earth is stationary and everything else is moving so the non-scientist would take a lot of convincing to get them to believe otherwise.
A: Others have already pointed out that heliocentrism was a theory in Ancient Greece. But the shortest answer to geocentrism's prevalence is that geocentric systems like Ptolemy's more accurately predicted planetary movements than heliocentric systems until Kepler discovered that the planets' orbits were elliptical and not perfect circles. You can't persuade anyone to your theory if you can't prove it or accurately relate your model to observation via accurate prediction. Ironically, geocentrism persisted because the ancients were decent scientists and were unimpressed by (pre-Kepler) heliocentrism's failure to support itself with observational data.
The Catholic Church had very little to do with the longevity of geocentric theory, as that theory predated the Catholic Church and Church views of the world were based on Roman and Greek science rather than the other way around. Also, once Kepler proposed the theory of elliptical orbits, heliocentrism became such a simple model compared to Ptolemy's unwieldly cycles and epicycles, that heliocentrism rapidly gained in popularity and quickly became the dominant theory.
A: To build on dagorym's answer, the Greeks did try to measure Earth's motion around the Sun to show heliocentrism, but the closest stars were still too far away to show any parallax that would verify that theory.
