What version of Civ are you using, and patch level?
Well, if a scenario uses custom units, it will be included with the download. Did you check that thoroughly to see if they are there? You can also post a comment for the scenario if it has a thread on CFC.
Also, in do you know how to get to the unit files? (C:/Program Files/Infrogames Interactive/Civilization III/Scenarios/Scenario Name/Art/Units - for Vanilla civ. If you use PtW, add "/Play the World" in between CivIII and Scenarios, or add "Conquests" in between CivIII and Scenarios. Did you get there ok?