Well there will be dirt in the boxes so even if you did set them up on bricks the dirt and moisture from the dirt would still be touching it.
I'm not completely zapping down your idea, but maybe containers like the 5 gallon buckets would be better....
Lets see working with your idea of using cardboard boxes.... thicker the cardboard and even waxed boxes would hold up longer. What is your idea with using the boxes, is it to keep the dirt mounded up? is it because the boxes themselves will break down eventually?
what about drainaige? You could put holes in the bottom and even the sides if you wanted to.
They have the little peat pots that plants are started in... they are then planted in the soil, the peat breaks down and you don't diturb the roots, so using the cardoard... but on a much larger scale for the outside edges of the garden I could see that working. With the little peat pots you have to worry about them drying out quickly, don't know if it's the material of the peat that wicks the water away, the fact that the amount of soil is usually small, the soil itself is more designed to start seeds not grow them long term... it would be doable, though, better soil, something a little heavier...
you could make the cardboard thicker... stacking a slightly smaller box inside a bigger one, maybe even doing this a couple of times to make the walls thicker.
what will happen though once the boxes do break down, would you just get new boxes and fill the soil back into them?
is the driveway concrete or gravel? I wonder if the boxes/dirt would stain the concrete? would your landlady mind if it did? I guess it could be cleaned somehow but you might loose some of your deposit eventually.
I think it's neat you are thinking outside the box!!!!
me and my play on words