I don't live in the States, so I'm probably not the best person to answer this question. I hope somebody better informed than I am will jump in. In the meanwhile, these are things I would probably be doing if I were in your position.
1) Check out my State laws, and find out how they would impact my poly family. If they are extremely anti-poly (e.g. I believe that in some States it is illegal for more than 2 unrelated adults to co-habit), then I would consider moving to a more liberal State. I know that's not easy, but it is probably easier to move in advance than to live through the kind of strain that the Browns went through.
2) Check out the school my children attend and try to find out how accommodating they are towards minority groups in general. In particular, if they are supportive of LGBT people, they might be more likely to be supportive of poly, too. On the other hand, if I lived in the Anoka-Hennepin school district or somewhere similar, I would take absolutely no risks with what my children might say in school.
3) In either case, so that my kids won't think poly is some weird thing that just I do (because they *will* figure it out at some point, and who needs a judgmental teen), I would talk with them in advance about marriage and relationship traditions in other cultures and in history. Polygyny, as well as concubinage, was common in the Old Testament. Polygyny is still practiced by many Muslims today, as well as in some older traditional cultures, such as some South African groups. Polyandry is still practiced by some groups in Tibet. And the Mosuo in China are a strictly matriarchal culture that practices 'walking marriages' where people don't move out of their mothers' houses, where women with children rely more on their siblings for support than on the children's fathers, and where couples do not live together. All this will help to put your relationship into a global cultural context for them, so it won't seem so alien. At least, it worked for my kids! You can get some beautifully illustrated geography/religion books for children which cover various religious customs, and these would include marriage customs. Usborne Books is likely to have something worthwhile along those lines, and perhaps Dorling Kindersley. It should also be possible to get short children's books on individual cultures from a child's point of view which would cover this. Global Citizenship is your friend!
4) In extreme circumstances, it is possible to tell children, friends and neighbours that a partner is your 'friend' or 'housemate', but please do not underestimate how hurtful this will be for your incoming partner to live with in the long term. If this is how it has to be where you currently live, do make concrete plans for the future (all together) that involve 'coming out' at some point, and take definite steps towards it.
5) Children are pretty perceptive, and may get worried about the stability of their parents' relationship if they pick up on the affection between a parent and a 'friend', or even a partner (if they know that you are all partners). They need to see open physical affection - such as hugs - between their parents to reassure them, and also affection - even if not physical - and respect between *all* of the people in the poly relationship to reassure them that everyone is really alright with one another. Of course, this is important for the children's own well-being, but also their stress may noticed at school, and they may be coaxed to talk about the 'problems they're having at home.' For the same reasons, it's important for *all* partners to work at their relationships with the children, and for the children not to feel that their wants and needs are regularly overlooked because the adults in their home are too busy processing relationship issues to pay attention to them.
Well, that's all I can think of for now, and it's turned out a lot more prescriptive than I meant it to, but I hope some of it may help you all.