Switch to a fertilizer that does not contain nitrogen or cut way back on the amount of fertilizer you use. Without this source of nitrogen, your dog's urine might not cause the grass to die.
Leave out more water for your dog and leave a dish of water outside when your dog's in the yard. The more water your dog consumes, the more diluted the urine will be, which will stop the urine from killing the grass.
Change your dog's food. Dog foods that contain less waste products, such as meat byproducts, are easier for your dog to digest and will cause less nitrogen in the urine.
Water down the area of your lawn where your dog frequently urinates more often. Watering right after your dog urinates can rinse away the urine and stop the grass from dying.
Give your dog a patch of soft soil, such as potting soil, or sand in the yard. Praise your dog when urinating in this area, rather than on your grass.