I'm aware that it's a no-no to post the same question on multiple Stack Exchange sites. I'm curious, however: what do people feel about posting the same question on non-SE sites, such as MetaOptimize and AnalyticBridge? These seem to be largely disjoint communities, so I don't think I'm bothering the same people more than once. And if I get a good answer on one, I would plan to cross-post it to another so everyone benefits. Still, I'm not sure what the community norms are for this.
Depending on the site, you may want to consider sharing a link to the Stack Exchange question, instead of completely cross-posting it. In essence, this is less of a "Here's my question, help me out", and more of a "Hey, I'm part of this statistics Q&A over here, it's quite nice and you should try it out. (Oh, and help me)" type of action. This is less of an option for Q&A sites that operate similar to ours (such as MetaOptimize), but for forums and other communities that you don't have discrete "questions", then it is a viable route.
Basically, promoting the site as a whole to attract new users to the site, while also attracting people who might be able to answer your question. It's also a pretty helpful thing to do - by sharing a question in need to those who can address it, you'll be attracting experts to the site who can assist, rather than spectators and other net-wanderers.
There are also a set of badges you can earn if you use special links. If you click
link beneath the tags of your question (next to things like
flag, you will get a special link that has your UserID associated with it. By sharing that with others, you can get badges like Announcer that reward the promotion of sites, all while still potentially getting an answer to your question.
Cross-posting remains a valid options, but as Jeff recommends, make sure to stay on top of the developments on the sites. You don't want people of any site to end up wasting time providing an identical answer that you had gotten a couple days earlier from a different site altogether.
And if I get a good answer on one, I would plan to cross-post it to another so everyone benefits. Still I'm not sure what the community norms are for this.
That is indeed the one requirement I have on this, so it seems you are on the right track.
Post a question to five different (non-se!) sites on the internet if you must, but you are now morally obligated to be responsible for the care and feeding of all five of those questions -- anything less is disrespectful to the audiences of those five sites.