Thanks for the input, please keep it coming.
For some additional context (I should have included this in the original post)...
We've been to Japan before (4 or 5 times?), so we've hit many of the typical first-visit-tourist-circuit places. Previous trips were during Fall, Summer and Winter, this will be our first time there in the Spring (though late Spring down in the south). We've always traveled independently, we have found that Japan is incredibly easy and always delightful for us, we expect to return often. We've been all over Hokkaido (absolutely wonderful), and much (but not all) of Honshu. This will be our first visit to Southern Japan (and our first Japan trip that does not include a stop in Tokyo).
I have a solid list of many (most? all?) of Kyushu's most obvious tourist attractions - from Nagasaki and the multiple A-Bomb-associated sites, to some of the nature-oriented locations (Aso, Beppu, etc.) with MAYBE a few of the kitschy, "crazy Japan" themed places tossed in. I know we will be late for sakura, that's OK (this is the south, so way late for that); we're actually hoping to catch the tail end of wisteria season.
As tempting as they are, we have made the choice to NOT try and get to any of the outlying and far-southern islands on this trip (believe me, this was a long and difficult negotiation); we will make a separate trip (maybe several) for those smaller islands - there's just too much stuff to see/do on Kyushu itself. So much, in fact, that we are still keeping open the possibility that we can't do all of Kyushu on this trip, and we may just do the "top half" and come back for the southern half - but I'm not ready to yield to that option yet, I'm still hoping for a complete circumnavigation...well see, I need to make some high-level decisions in the next few weeks.
We are flying in to and out from Fukuoka, and that will be our initial base - but we will have other bases around the island (most likely candidates to make the cut right now, in/near: Nagasaki, Kumamoto, Kagoshima, Miyazaki, Oita...though a couple of those may be one-night stands as we blow through). We will be staying at a mix of accommodations: some large international hotels in the bigger cities (staying for "free" with points, helps with the budget), some traditional ryokans, some onsens. (The place in Kikuchi looks great, thanks for the multiple pointers!). I'll put the Hello Kitty theme park on the "optional" list...
We will be there immediately AFTER Golden Week (we arrive the day after it ends, a conscious choice).
Our interests are fairly predictable (up to a point): we like culture, history, scenery, food, prefer to avoid crowds but can manage when needed. We like empty, natural places and odd human-made things. Not big fans of super-touristy attractions (overt tourist traps) but will chuckle and go along occasionally (Japanese tourist traps are so over-the-top, they're often good for a few laughs, eg that Hello Kitty theme park, and the sadly-now-closed Robot Restaurant in Tokyo....that was pretty trippy).
Please continue with any suggestions and recommendations, they are much appreciated. I'll be up to my armpits in planning for the next few weeks.
Arigato!