We were taught in school that when followed by a noun starting with a vowel sound, "the" is pronounced "thee". In other cases, it's pronounced "thuh" (aside from the emphatic "the").
I have noticed that some Americans say "thuh only" and "thuh hour" (these are the only examples I could find off the top of my head). Is this just a regional accent thing or are there exceptions to the simple rule I was taught in class?