I live abroad and this is something I am really conscious about as I bring up my kids. We read loads in English, and as we go through the year I try to link in the British traditions to put them on a par with the local ones, because my kids are so rooted in this tradition that the British side of their background are, despite my best efforts, rather alien/exotic to them. I'm even trying to teach them the clapping games we played in the playground (any www links, anyone - I was never very good at them) and the "Snow, Rain, Thunder, Lightning" handstand game 
I, too am an atheist but also at Christmas and Easter read my kids Bible stories, sing carols, etc. so that they get the basics. We live in a very Catholic country and they went to a Catholic preschool but despite that, one Easter when I asked them what their teacher - a nun - had told them about the Easter story, they said "She says the Easter bunny will come and hide chocolate eggs on the lawn." 
Also among the songs and rhymes I try to teach them are the less specifically "Christian" hymns I remember from school like "One more step along the world I go", "Autumn days when the grass is jewelled" - anyone remember that?. Had forgotten "When I needed a neighbour" - thanks, will bring into the repertoire.
But then there is another aspect that I also have to address - with this being a Catholic country, RE/RS at school is basically catechism and preparing for 1st Communion. There is no interfaith/other faith stuff at all, and this is also something I desparately try to rectify. Luckily our city has a slowly reviving Jewish quarter so we go to Jewish-themed workshops occasionally, but non white-Caucasian faces on the streets/in books are practically non-existent, as are styles of dress other than typically western.
Well, off to buy my pumpkin and fireworks (told them the story of Guy Fawkes' Night for the first time this year
. And tried to reconcile it with the ethical implications of burning an effigy of a human at a stake!!! [tying myself up in knots emoticon])