I would recommend drying like the Baron says: try laying them out flat, put some sort of cloth on the non-printed side to discharge the water (bin liners on both side will trap the water, thus making evaporation impossible, try putting a little pressure on it to prevent curling (eg. multiplex with a weight?). That way you will press out the water and make the structure of the cardboard compact again.
However, you boxes will surely never be like they once were.
Finally, never ever put cardboard items in non-heated/humid rooms! Cardboard will soak up water from humid air. This can result in fungus and other nasty things inside the cardboard.
Same goes for IKEA furniture, but makes sense, because they are made from laminated cardboard.
