In brief, I see no reason why this this couldn't work...with a few caveats.
Eclipses are very localized events, usually clear from only small portion of the Earth. Eclipse cycles also run on cycles, but these cycles are not monthly because the near monthly lunar cycle is one rotation of the moon around the Earth.
Eclipses result from specific alignments of the Sun-Moon-Earth and this cycle is about 18 years. Multiple eclipses can occur over this time period, but the Eclipses will not be in the same location, nor at the exact same time in the cycle. This is because of two factors; the eccentricity of orbit (orbits are not circular and predicting the alignment of elliptical paths is more complicated than predicting the alignment of circular paths), and the precession of the Earth-Moon system. This means changing these two things in the right way could get the desired effects.
In summary if the Earth's orbit about the sun, and the Moons orbit about the Earth were more circular, and if the Earth-Moon system did not precess, you could in theory get a system were an eclipse happened at consistent intervals and even perhaps with the eclipse happening once in a lunar cycle. But then come the caveats...lunar eclipses would only be view-able from one side of the Earth, and Solar eclipses only view-able from a specific locality on Earth, everyone else would not experience these events, so they would have to be cultural significant based upon locality. Also a lack of precession in both the Earth and Moon and a circular lunar orbit would affect climate, and rather severely. A lack of seasons and no tide cycle would become a problem and uneven temperatures would result in rather extreme weather. Of course, this could all be downplayed, but the climate differences would be an important consequence of having getting the desired eclipse cycle.