In Animal Crossing: New Leaf, grass can be worn away if you walk over it frequently. If you take the same path when you walk and run around your town, you will eventually create dirt paths in those areas. This page has more details about grass deterioration and growth.
Grass will wear away if you walk over it frequently. The rate of grass deterioration is unknown, but it wears away much more slowly than in Animal Crossing: City Folk. In City Folk, grass paths would start to show after a week. It takes much longer in New Leaf.
Grass gradually grows back. Grass will grow more quickly if there are lots of trees and flowers in your town. Rain and snow also help the grass to grow back.
Patterns on the Ground
Patterns on the ground do NOT protect against grass wear or influence grass wear in any way.