For years the residence of our street have been paying €100 each per year to a contractor to have the grass cut in in our street because the council would not do it. I assume that now we are paying property tax the council will take over this responsibility.
If the council are not responsible for cutting the grass in your area now I don't see how the introduction of the household charge/property tax would change anything.
My estate was taken in charge by the council this year (estate built in 1996). The council have stated that they could take over the grass cutting, however it might only be cut twice a year rather than once every 3 weeks or so by the private contractors hired by the management company.