That's something to think about although all the drip irrigation I have seen is above ground and I was leaning towards having everything under the soil to lessen the amount that would just evaporate. So basically I would drill a hole in the side of the box and have pipes running in kind of a U shape to water the whole bed. This also means the animals that will, despite my best efforts, somehow get in can't wreck the system too badly. Edit: It seems like what I'm getting at is essentially a drip irrigation system just without the actual dripping since it's underground. "sub-surface drip"
Look into drip tape. You can cover it and it still works fine. You will have to have some sort of hole in your box but it could be as small as 1/4 in for the small drip tape. The 1/2 in stuff can only be run in straight lines but the 1/4 in stuff can be looped. It works a lot better than the cheap soaker hoses you buy at Walmart. Drip irrigation doesn't care if you burry it since you aren't relying on water pressure for dispersal