This either says nothing for my memory or food experiences....but in Dublin the only "must eat" food memories I have are attached to the scones sold at Avoca. Those are fantastic.
It's not that the food in Dublin is "bad", and if you're willing to spend money there are some really decent places...but not much I'd go out on a limb and say is "must have". Leo Burdocks is the more famous fish n chips place in the area if you're seeking that out.
My paltry advice might be because I was there for a year as a student and didn't do as much foodie stuff - but all in all, as much as I love Dublin - eating there was never the high point.
We are not really a food cart city opr country like I remember from my manhattan days. Saying that every Saturday in Temple Bar Meeting house Sq there is a food market and lots of places to eat and take away from. there is Japanese, Crepes, veggie cornish pasties, Mexican (I like his sauces and he is from Mexico) and lovely olives and cheeses. you can make a great picnic or snack for later. There are also oysters fresh, venison burgers cooed to order and the Gallic Kitchen has lovely quiches and potato rostie cakes.
Also Enotecca della Langhe and Bar Italia in Bloom Quater North dise just over the millenium bridge. The Enotecca is more snacks and antipasti plates with wines mostly from Piedmonte and the later is a restaurant and reasonably prices more Roman style food.