Grass will wear away if you run over it. To prevent this you should walk everywhere or only run on patterns you place on the ground and stone pavements (like the one in the town plaza).
Patterns can be placed on the ground by using the design menu on the touch screen, just like you can use custom designs as clothes.
You can regrow grass at a faster rate by planting trees and flowers on or around the dirt. Be sure to water the flowers to help the grass grow.