If you time travel too much or you aren't taking care of the environment, your grass will start to wear away. The grass is the green on the ground. Don't get it mixed up with weeds or flowers. Here are a few ways I know that grow grass back.

Minor time traveling
Turn on your game and right when you step outside, save and quit. Then change the date to a day forward, step outside, save and quit again. Repeat and your grass should be growing back.

Plants
Planting flowers and trees is very helpful. When you plant a tree, the grass will go around and under it. Then when you cut it down the grass will stay. Same with flowers, except if you use gold roses it'll speed up the process.

Creating trails
Simple. Just make a trail with your patterns on where you walk to the most.

And for the impatient people, you can just get a grass design from Wendell if you give him a Black Bass. He'll appear on random weekdays after 6AM and midnight. Nothing will ever grow there, but it may be worth it to get the town looking good.