I've done pretty much exactly what you describe: training on a (really terrible old beat up MTB) and then using a rented road bike (totally different configuration, weight, etc.) for a century (RideLondon-Surrey 100) with no problems. I can't see how I wold have survived riding that distance (on those hills!) in my MTB. So I strongly recommend switching!
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The answers so far sound like good general advice, but it's not clear that any of those answering have actually tried what they are advising against.