Try Portstewart - it is 3 miles around the coast from Royal Portrush, and has THE best opening hole in all of golf. I guess you could call me bias - - - I was born and raised in Portstewart, and still call Portstewart and Portrush as my home courses. Portstewart has fallen in the shadows of Portrush for a long time, but with the addition of 7 new holes carved thru the dunes (designed by a club member if you will!), it now stands on it's own merits. Darren Clarke enjoys the golf there, and so will you.
If you don't feel like traveling up to the North, the par-73 County Louth Golf Club would be my next bet - it is situated in the town of Baltray, and is probably about a 90 minute drive from RCD. They play the East of Ireland Amateur Championship there every year, and it really falls under the radar with Americans - the members will treat you like their own, and the experience will always be a good one. Come to think of it, that just about describes every course in Ireland - the locals really enhance the overall experience..
P.S. If you decide on Portstewart, you can take a ferry to Ballyliffin - a neat experience for a visitor. It takes 40 minutes door to door from Portrush.