We do deep-cleans (replacing the litter + scrubbing out the box) twice a month. My process is basically:
1) Empty the old litter in a trash bag
2) Put the litter box in the bathtub and rinse it well with hot water
3) Scrub with diluted bleach-water, then rinse again
4) Put the box back and refill with fresh litter
Sometimes, if we're out of bleach, I use white vinegar instead. And I only use bleach in very small amounts, around a 1:20 ratio or so.
As for your situation, is there any way you could try to move the box(es?) somewhere that the dogs couldn't get to them? Clumping litter would make the whole multi-cat household thing a whole lot easier to manage.
Otherwise, rinsing them out in the tub is the only other thing I can think of.
Proud parent of Scamper (age 12) and Robin (age 4)
At the rainbow bridge: Slippy, Faramir, Gabriel, Lucky, Ammy