Box springs help evenly distribute pressure on the mattress not just while laying but also when you're getting on/off the bed resulting in concentrated pressure points. They are for the most part, very firm and yield only the slightest bit. Most box strings don't even consist of springs at all. They normally consist of cheap flexible wood, thin upholstery batting, and cardboard. Without a box spring you run a higher risk of popping a spring in your mattress. The thickness of a box spring is only for decoration while the functional properties could be reduced to a much smaller design.
Slats are typically used with mattresses that don't have springs (coils). If you want to go the slat approach and have a mattress with springs, I'd strongly suggest using a thin yet strong sheet like material over the slats. Something like plywood or faux wood paneling.
Take these concepts into mind whilst building your new frame and you should be perfectly fine without a traditional box spring.