lots of great biking.
wicklow mountains on the edge of dublin is probably the most popular place, a trail centre (Balinastoe) opened there last year, though the best riding is on trails built by locals so your best hooking up with one of the clubs. Very few marked bridleway type trails (except wicklow way which is great to ride) and biking isnt officially legal a lot of the time but you wont actually be stopped.
also trail centre near limerick, plenty of riding in the north and some in cork, galway etc.
good racing scene too if you’re interested in that.
check out
http://www.mtbireland.com
http://www.irishxcmtb.com
http://www.madmtb.com
http://www.epicmtb.com