What if you lined the boxes with a pond liner? It will prevent the water from seeping out. Then you'd have to be careful not to overwater your plants, as there will be no drainage holes.

You could line them, as faltered said, and also keep the plants in their pots. You could camoflage the tops (so that you can't tell they are still in their pots) with spanish moss or something of that nature. Don't use sphagnum moss, as it is already being overharvested.

OR, you could line them, making them completely water-tight, and then fill them with water and grow water plants! There are quite a few, for both sunny and shady exposures. I grew umbrella plant (cyperus) in my small pond on my balcony this year, and it's great! Nice and tall, dramatic spiky foliage, and its roots would fit nicely in a box the height and width you described. Others include elephant ear (colocasia and alocasia), juncus effusis (unicorn grass or corkscrew rush), cattails, floaters like water lettuce, water hyacinth, azolla, duckweed, etc., and many, many more. Even some common houseplants can be grown in nothing but water, like syngonium (arrow-head plant), some palms, wandering jew and ivy cuttings, etc. Don't be afraid to experiment! You would need to put some pea gravel in the boxes to stabilize the emergent plants, or just leave them in their pots. The best part is: no water or dirt messes, and you'd only have to top up the water occasionally (more on very hot days). You would have to empty them in the winter though, or the water would freeze and crack the planters. And pretty much all water plants are tropical (except cattails) and you could overwinter them inside as houseplants (but still growing in water). Some of them go dormant, like elephant ears and water lilies, so then all you'd need to do is store the tuber until next spring.