Staying off the grass is what worked for me. I created paths and stayed on the paths as much as possible, but it still takes a couple of weeks to a month for the grass to regrow. Rain and snow will also regrow the grass quicker so if your town is currently set during winter, you should start seeing the patches fill in each time it snows.
Of course, there are dirt patches that will never grow grass. They can be a real eyesore, but placing houses, PWPs, flowers, trees and bushes over them helps.
It is also possible to time travel to get the grass to regrow but since I don't time travel, I can't tell you the safest way to do that without losing villagers, etc. Hopefully someone else can fill you in on those specifics if you're interested.