Archimedes constructed a planetarium where as described by Cicero "he had thought out a way to represent accurately by a single device for turning the globe those various and divergent movements with their different rates of speed. And when Gallus moved the globe, it was actually true that the moon was always as many revolutions behind the sun on the bronze contrivance as would agree with the number of days it was behind in the sky. Thus the same eclipse of the sun happened on the globe as would actually happen..." (see sources).
When the Antikythera mechanism was found some suggested that similar gear system was used by Archimedes. However, the Antikythera mechanism (dated to 150-100 BC) has an epicyclic gear for the lunar motion, so under the conventional timeline there are two problems with this idea.
The first problem is Apollonius. In Almagest XII,1 Ptolemy credits him with proving theorems about epicycles, and this seems to be the earliest association of epicycles with anybody's name. Some biographers say that "the theory of epicycles certainly predates Apollonius", but I am not sure where this "certainly" comes from.
The second problem is Hipparchus. Even Apollonius is only credited with theorems, Hipparchus (again according to Ptolemy) was the first one to apply epicycles to lunar motion, and he did it based on Babylonian data, which Archimedes presumably didn't have a century earlier. By the way, Hipparchus lived around the time the Antikythera was made. However, a gear similar to Antikythera's, but dated to before Hipparchus (200-150 BC), was found in a shipwreck that can be linked directly to Syracuse, and hence to Archimedes.
On the other hand, if Archimedes did not use epicycles the only model available to him would have been of Eudoxian homocentric spheres. According to modern reconstructions (p.226) reproducing the uneven motion of the Moon even crudely (and predicting eclipses mentioned by Cicero) would require at least three spheres rotating at different speeds under different inclinations, and with outer spheres transferring motion to inner spheres to combine it with their internal rotations. Even more spheres are required to reproduce retrogressions of the planets.
Ptolemy is not exactly a great authority on historical matters (he managed to mention no astronomer between himself and Hipparchus, a span of over 200 years), and his attributions to Apollonius and Hipparchus were specifically questioned by some scholars. Life dates of Archimedes (287-212 BC) and Apollonius (262-190 BC) overlap, and we know that both corresponded with Alexandrians, at least two of which, Eratothenes and Aristarchus, performed astronomical measurements (Archimedes wrote to Eratothenes and mentioned Aristarchus in Sand Reckoner). Finally, some arrangements of Eudoxian spheres are suggestive (p. 227) of the simpler epicyclic scheme.
Is it plausible in the light of recent developments that Archimedes/Alexandrians already developed epicyclic models of the Moon (and possibly planets) a century before Hipparchus? Alternatively, is there a mechanical arrangement of multiple nested spheres with motion transfer combined with internal rotations, that uses gears and can be powered from a single rotational driving source (within Archimedes's capabilities)? Modern reconstructions of the planetarium that I've seen do not seem to be up to the task.