Hi Gringo1000,
You have to pronounce the "u" in -uir unless it is preceded by the letter g. The only examples that I can think of this latter case are the verbs seguir, conseguir, and proseguir, but there are surely others. The rest will preserve the "u": construir, concluir, huir.
The reason for this is that the "u" in these cases transforms the g from the "j" sound (approximating the English "h") as in gira and corregir, to the sound that approximates the English "g" (as in gallo).
To learn more about the pronunciation of vowel combinations you should look up a good resource on dipthongs (diptongos) in Spanish, and the difference between strong and weak vowels.