If you want grass and not to see the stones, then install a channel somehow (curb) to send the water toward a catchment basin, then tile down the hill to the low area where it can dump out. Hopefully not onto your neighbors property. You can't stop normal rainfall and runoff, it is natural, but if you channel it toward his house you might have some liability.
You have several options depending on what you want the yard to look like and how much labor or ($$$) you want to put into it.
An option is pipe. The black HDPE (high density polyethylene) pipe that comes on a roll and is sold by many big box home improvement stores. They also sell pre-fabricated 'catch basins' made to install on one end of the pipe to collect the water. The basin can be installed on the 'driveway' side of your curb or (earthen) berm to catch the water that's stopped by the curb/berm, and direct it to a pipe. You already have the 'ruts' you spoke of. A little effort on your part to excavate those ruts more, to about 16" to 18", will allow for installation of the pipe across your yard to a point of discharge where you choose, lower than the driveway. The big box outlets (especially HD, the orange one) are pretty good at providing instructions on how to put things together. And it shouldn't cost an arm and a leg, either.