Opened windows along with box fans (or better yet, a house fan: http://en.wikipedia.org/wiki/Whole-house_fan) will help, but you will still be in a worse situation than if you had a house designed to stay cool and the same fans.

Houses designed to stay cool without AC, among other things, tend to have more intelligent window placement, and windows that open much wider than is perhaps normal (you won't find those rubbish casement windows that only crank open to an acute angle in homes designed to stay cool without AC: http://www.choicesil.com/images/House1-16-08%20017.jpg).

Given the option of staying in a house that never had AC, or a house with broken AC, I would chose the former every time. Fans and open windows will help in either case, but they will be more effective in the home that prioritized that usecase.