I am an Atheist. My partner is also an Atheist, and Jewish and he has two young daughters. I am American and not Jewish. My partner was born and raised in Russia (at the time USSR), immigrated to, and lived in Israel for 13 years, and is now in the States.
I say he is Jewish because even though he does not believe in God, he identifies with being Jewish on a cultural and ethnic level. There is an undeniable truth to the cultural aspect of Jewish identity.
His children's mother, also a Russian Jew who spent some time in Israel, is a little religious.
My partner's oldest daughter, who is in 1st grade, strongly identifies with being Jewish, an identity that she seems to positively reinforce herself in the context of Midwest, and very Christian, America. She actively seeks out Jewish "things" and Jewish people.
I have had periods of strong anti-religiosity, though I feel my attitude has mild and I have been faced with the interesting choice of whether to practice traditional Jewish holidays in our household (which is ironic because I view the God of the old testament to be monstrous). We certainly do not follow Kosher laws, though the children's mother maintains these holidays and kosher laws somewhat liberally. I am considering maintaining and fostering the children's Jewish identity and culture by doing Jewish holidays. Their father wants them to understand and know Jewish history. There seems to be something valuable in the practice of ritual.
Though, with this said, I am Atheist. The children know that both me and their father do not believe in God. We have explained what it means to be Atheist but we have not told the children whether they should believe in God or not. I think it is important to have the children make their own choice on this matter, and be able to think about it critically.
It would be interesting to hear other people's reactions on this issue.