Is there a way to get rid of dirt patches?

In all Animal Crossing games, grass deteriorates from a player character walking over an area, so simply not walking on a dirt area at all will allow it to very slowly regrow grass. In New Leaf, this regrow happens in real-time, so time when you're not in the game counts as well.

The rate of regrowth can be doubled by planting flowers on the patch, and watering those plants everyday. However, if you walkover those flowers then the effect will be nullified and the grass will continue to deteriorate. A similar effect occurs with trees, bushes, and even precipitation.

So in summary:

• Don't walk on the deteriorating grass, or patch of dirt, you are trying to remove
• Plants flowers, trees, and bushes on the affected area
• If you plant flowers, water them everyday
• Precipitation counts as well

You can plant flowers over the spots, and then water them every day. Eventually, the grass will replenish itself if you continue to water your flowers over it!