There were times when I thought I had lost the battle for Irishness but one day DS said 'We don't talk like other people', and when pressed, he said it was hard to describe but I kept on pressing so he tried, and the gist of his answer was that we used different words and phrases, longer words, whole sentences in conversation, had a different sense of humour, didn't shy off a good argument, shared different sorts of things with each other than his friends and his parents did - I think what it boiled down to was a different family culture; although the DCs' accents are not Irish in the least, they could fit in fast in Ireland because they listen to different things in speech and take conversations in a different direction than American peers.
I never wanted them feeling they were outsiders, however, so I didn't labour the point of Irishness, and the fake Oirisheyness in the US was something I really wanted to avoid. I tried introducing them to Marmite (no joy), blackcurrant flavoured this and that (they loved), The Clangers on YouTube (love at first sight), and the Eurovision (lead balloon), but we normally had American Christmas desserts as well as something I loved as a child (meringues and a buche de noel) and did Thanksgiving in full American style. I see it more as sharing an important part of myself with them than making them Half-anything, identity wise - I also try to share my love of reading, music, history, maps, art, architecture, design, etc.
One thing I hope they will grow up with is a sense of curiosity and the awareness that there is more to the world than your own 'backyard'. Another thing I think is important to me is that they know as much as possible of their family history, and I have come to realise in sharing that with them the importance of the family farm to me, and the sense of being rooted somewhere. On my dad's side, the family were globetrotters who spent a few generations on one continent or another and then returned to a different part of Ireland, but mum's family always had the farm, or some farm, in one particular corner of the countryside, and although the DCs are two generations removed from the land at this point, I hope they will acquire a sense that that is a little patch of the earth that they have a deep connection to.