what you see up top is the organisms attempt to reproduce. you can lop off the tops before the gills open (the spores are released when the gills open) and you can also dig them out...the part that's underground can be pretty extensive though, so, that will help some. it won't completely resolve the issue tho.
the best defense is a good offense in this case. what that means is maintaining the lawn properly so that it's as healthy as it can be. and. most important is to keep on cleaning up the dogs droppings - poo is prime real estate in a mold spore's eyes!
spores can lay dormant for years and they can also blow in on the wind, so, you'll never be completely without them. you can keep them to a minimum, though.
remove them as they appear and continue doing so and keep the area clear of decaying matter (leaves, poo, wood chunks, etc).
i would take a few (either the actual mushroom or some good/clear pics) to your local county extension office for id. if they are the poisonous type, you definitely want to know so you can keep the dogs away from them until you have time to do some digging.