I will be in Dublin on Holiday for 6 days in May. Rick's 2012 guide plans out 3 days. Are there any day trips you would recommend for the other 3 days? I will not have a car. FYI 2 of the 6 days are a Sunday and a Monday so that may complicate access to sites and museums.
How about taking the train from Dublin to Belfast? While there may be a few things closed on a Sunday, you can always take a Black Cab Tour, hop on/hop off bus, or get a city bus to take you over to see the murals along The Falls Road and/or in the Shankill. Plenty in Belfast to keep you busy for a day.
There are several companies that offer bus tours to Glendalough/Powerscourt Gardens/Wicklow and Newgrange/Boyne Valley, including Bus Eireann, Ireland's national bus company. This page on the Bus Eireann site has links to detailed information.
A trip to Belfast on a sunday would be ideal. I took the black cab trip and with no traffic it was a great way to see Belfast. The drivers are trained to give lots of information. NewGrange would be another good day trip.
If you take the train to Belfast and back to Dublin, be aware that on Sundays the first train doesn't leave Dublin until 10:00 and the latest you can leave Belfast is 19:00. Taking the bus will work better on a Sunday. On weekdays and Saturdays you can leave Dublin as early as 07:35 and leave Belfast as late as 20:10.