What I do is place the run in a spot that the dog has not piddled on, leave it there till the next move and periodically throw buckets of water on the spot she has piddled on if we have not had any rain for a while to wash the grass off. I also put water on the spot where the guinea pigs run has been after I move it as of course they will have piddled there too and they wont want to eat where they have wee'd so if I manage to move the run back in the space of a week I just think its nice for them to have fresh grass again.
Just try to wash the spot the dog has been - both wee and poo, a watering can will suffice, after you have seen the dog go (and picked up), this should keep your grass nicer for them to eat.