Via feministing, here's a story about a couple in Sweden who's raising their child without gender pronouns or gendered behavior. In fact, they're keeping their child's sex a secret from the world at large in hopes that the child will be treated as an individual instead of a man or a woman.
Pop's parents [see footnote], both 24, made a decision when their baby was born to keep Pop's sex a secret. Aside from a select few - those who have changed the child's diaper - nobody knows Pop's gender; if anyone enquires, Pop's parents simply say they don't disclose this information.
In an interview with newspaper Svenska Dagbladet in March, the parents were quoted saying their decision was rooted in the feminist philosophy that gender is a social construction.
"We want Pop to grow up more freely and avoid being forced into a specific gender mould from the outset," Pop's mother said. "It's cruel to bring a child into the world with a blue or pink stamp on their forehead."
The child's parents said so long as they keep Pop's gender a secret, he or she will be able to avoid preconceived notions of how people should be treated if male or female.
It's an interesting idea, and despite what the gender fascist quoted in the article says, it won't hurt Pop to not be forced into a box when it comes to gender. If she's correct that kids naturally gravitate towards either the male or female gender, then there'll be no problem here - Pop will decide for Pop-self if Pop is a boy or a girl at the age when Pop is naturally drawn to that gender.
It doesn't seem like the parents are trying to raise "Pop" to be gender-free; it seems like they just recognize that it can be limiting to tell another person whether they are male or female. I've read enough from the transgender contributors on TBP to understand that being one gender is, on some level, hard-wired into people (something that as a cis-man I don't have to think about too much since I was socialized as a male, have male body, and strongly identify as male).
That doesn't mean that everything that comes with being male or female (as in, that which is a result of being treated in accordance with one's apparent gender) is, in fact, hard-wired. And as Ann points out, it's something that the kid will eventually have to choose as society pressures rather forcefully people into the binary. Not that there's an issue with that. Pop will at least be free to declare him/herself as a boy or a girl, and there's nothing wrong with that.
Ultimately, it doesn't seem like a bad idea to just treat a child as if he/she weere a human being instead of a boy or a girl. And whatever bad side people can bring up (that this child will eventually have to choose or that this child will show characteristics of male or female eventually), it seems like they're inevitabilities anyway. More power to the parents of Pop, even though their experiment will probably be ruined by the time Pop starts school and will feel pressured to choose.
I don't agree with the parents that gender is solely a social construct, but if it isn't there doesn't seem to be any reason to be so insecure as to force one gender or the other on a child. These parents sound involved enough that we don't have to worry about this kid, and nothing else should really matter.