The best (or at least most convincing) publicity is usually word of mouth, so I would say that members of those organizations should just post something on their forums or whatever. Following any rules about publicizing other sites, of course.
For the larger organizations like AOPA I suppose an email to their news editor might be worth a shot, provided that it's well written and 'sells' SE effectively. "Q&A network StackExchange launches aviation site" could be good filler material for them. The only issue with that is establishing who actually represents SE and/or aviation.SE to the outside world? I don't know if there's already an established process or precedent for SE in general?
I'm an AOPA member and although I'm not active on their forums I considered posting something there along these lines:
If you already use any of the StackExchange Q&A sites you might
like to know that they now have an aviation site in public beta.
If you've never come across them you can read more about them
here, but basically it's a bunch of Q&A sites on a lot of
different (often technical) topics; their IT-related sites in
particular are very well known. They focus on answers, not discussions,
so the 'feel' of the site is a bit different from here.
The main reason I haven't done it yet is that I have never posted on their forums so posting a "check out this cool new site" message might look a bit like spam, especially to people unfamiliar with SE. A post like that certainly wouldn't be considered useful here. On the other hand, only AOPA members can post there, so at least they'd know I'm not a bot!
EDIT: I just discovered that this point is addressed in the "7 essential questions" meta post. This suggestion is very interesting:
sharing links to great questions and answers is the best way to start
I think that sounds like a great way to introduce the site on other forums, and I'd never even realized that there are badges for doing it. I guess that a posting about an interesting question is likely to be much better received than a typical 'check this out' post.