Public Grievance in Evony is a number whose purpose is to stymie the growth of Loyalty.
It does not do anything but act as a "filler" of numbers where you could instead have higher Loyalty or Tax Rate. It is affected by losing attacks to your city, as well as Disaster Relief and Praying.
If your City Loyalty/Grievance is in a very bad way, due to poor tax management, overworking or being under heavy attack, a miracle cure exists in the form of the Speech Text item.
Thanks to the amazing powers of good oratory, this instantly sets your Loyalty to 100 and your Grievance to 0. Such fortune doesn't come cheap though - it's 100 Game Cents in the Shop. If the alternative is losing your main city that may seem a small price to pay.