In reply to Moggsy: What are you looking for exactly? Big cross country/all mountain type spins across the mountains, little jaunts around the forest, perfectly kept trail centre trails?
The official situation at the moment, and for the last eternity too, is that mtbing is technically banned on Coilte (that be equivalent to forestry service to you) land. With the exception of a number of smallish trail centres dotted around the country. Cycling on other mountain land is often in a grey area too. However this is only the official situation in real life things are quite different. Dublin and Wicklow mountains are peppered with trails that get a fair bit of use most heavily used areas are Djouce Woods and Ticknock/3 Rock (which also has a small official trail). No one's ever been done, yet, for mtbing on these technically banned areas. Though because they're technically banned it's often difficult to find information about specific trails as mostly trail knowledge is passed through word of mouth or by just going out and exploring in areas you know there happen to be trails.
As previously mentioned there are decent official trails near Limerick with the Ballyhoura trail centre which is good but slightly overhyped in my opinion. In Dublin/Wicklow official areas are Ballinastoe Woods and Ticknock with official trails being in both instances OK but nothing special. If you want to go further afield best bet is get yourself an OS map for the area you're thinking of and then go exploring. Or if you're looking for something more detailed try asking on any of the local club forums (madmtb, epicmtb are the main clubs in dublin or google to find details of more regional clubs) you'll generally get invited on a spin or something like that.