As far as I know there is just one Orange march each year in the Republic, at Rossnowlagh in Co. Donegal. Orange lodges from Dublin and the border areas like Cavan and Monaghan usually go North to march.
A year or two back the Mayor of Dublin thought it would be a fine gesture to invite the Orange order to march in Dublin, like they used to do in pre-independence days. The thinking was they wouldn't be any threat to Dubliners and therefore would be a quaint spectacle for the citizenry.
Delighted Orangemen observed how strange it was they were being invited to march in Dublin but couldn't do so in parts of the North. This caused Sinn Fein and others to vehemently oppose the invitation for fear of handing the bowler-hatted ones a propaganda coup while Drumcree was still on the boil. Violence was threatened and the invite was soon withdrawn.
By the way, McGrath, I remember John Hume many years ago also looking forward to the day when the 12th would become a harmless cultural event.