Have there ever been any challenges to this commonly-held belief in the past? And if so, what sorts of examples were used as supporting evidence? Were there other cultures that supported this hypothesis with evidence before the Europeans?

I am aware of one example in the hypotheses of Aristarchus of Samos, if a European example fits into what you are asking. While his actual works have not survived, Archimedes does make reference to a theory he held which supported the idea of stationary stars and a stationary sun at the center of the universe, as well as a universe multiple times more expansive than previously believed.^1 Given technological limitations of the time, much of this is based upon assumption, given a lack of observable stellar parallax (which can't be observed without a telescope anyway).

This is the only example of pre-Copernican heliocentric theory of which I am aware, though I find it more suitable to defer to others on early non-European astronomy.

1. Archimedes references the work The Sand Reckoner, in which this hypothesis was supposed to have been found.

As /u/turtleeatingalderman points out, Aristarchus is one of the original proposers (later picked up by Copernicus), but Plutarch also references Seleucus of Babylon (or Seleucia), a 2nd century BC astronomer who also held the view. It depends on the level of what classifies as heliocentrism, but Hecalides of Pontus, Hicetas, Archimedes, and Ecphantus all said the earth moved or adopted the hypothesis (or considered the hypothesis as plausible). Pliny, Seneca, and Vitruvius possibly might have thought that the planets moved too, but that's mostly down to interpreting things between the lines, rather than an outright espousal of heliocentrism.

Things pretty much died down about heliocentrism until Copernicus, primarily because Ptolemy's astronomy superseded the heliocentric view, but also because the texts weren't available until then. There's also the fact that 'science' was hindered by the wars between Greece and Rome, which severely hindered Greek scientific thought - Archimedes was killed, cities reduced to slavery, Carthage and Corinth were razed, and Ptolemy VIII persecuted the Greeks wherever he could - Alexandria was depopulated of Greeks according to Polybius, so that put a stop to Greek scientific endeavors. The Romans were interested in science, but tended to rely on the Greeks - whom they had been slaughtering and enslaving, so it took some generations for science to be picked up again.

If you want more reading on this, the best text is Lindberg's The Beginnings of Western Science. Just be sure you have the latest edition so you get the extra material on non-Western topics.