We lived in Puerto Rico when I was a young girl, and our house didn't have AC. However, the houses were designed to take advantage of the natural breezes on a Caribbean island, we had lots of mature trees to shade the property, and the floors were terrazo tile- nice and cool on the feet. We never suffered in the island heat- just opened up the windows to let the breeze through. We spent most of our time on our big covered porch- it was our outdoor living room.
I live in not-so-tropical North Carolina now, but couldn't live without AC!! Of course, we live in a house with no tall trees around it, carpeted floors, and without those nice island tradewinds.