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Castlevania III in the US uses the MMC-5 mapper[0] which is relatively complicated as far as mappers go. Most modern emulators have no issue with it, but many cheap Nintendo-on-a-Chip clones have it[1]. Castlevania III in Japan used expansion audio through connectors that were not available in the US, and thus has enhanced sound compared to the US counterpart.

Emulation systems, such as the Retron 5, deal with Castlevania III with no issue.

[0] http://wiki.nesdev.com/w/index.php/MMC5

[1] To see a common NES clone struggle and die in the process of playing Castlevania III, see this video from Satoshi Matrix: https://youtu.be/OnPv1xHsbvg?t=13m56s