I've found that planting flowers seem to work best. Some areas are just naturally barren, and nothing will make those areas grassy. Ever. Wish they'd get rid of that little mechanic, though...
Are they dirt patches that were there originally or from grass deterioration? If the patches were from deterioration, then plant flowers there and water them. If the patches have always been there, then unfortunately they're permanent.
If you want you can come over to my town and dirt-patch it up for me to get over your frustration.
*Thankfully* grass grows back rather quickly in this game, foliage or no foliage -- just takes time. However, as the others have mentioned plants surrounding and on the area speeds up the process.
Then the patches should go away soon. Like others said, just plant some flowers on it and water them everyday. And obviously don't run on those spots >.<
Rain and snow helps tremendously, too. Last week in my town, it has been snowing a lot, and dirt patches that were left from moved-out villagers started growing back quite generously.
Definitely enjoy the weather we've been having in game! I had read that the grass grows back faster in winter. I had my doubts, but I've realized that it's true! Just plant flowers over the dirt like others said, and with a few days of snow, you'll have your grass back in no time!