I'm having some stairs fitted at the moment that only have wood on the treads and no the riders, I really like that look. Feels a little more modern than a full wood staircase, also cheaper ;-)
I'll take another pic in a couple of days when it's done. It should look something like this when it's done:
They screwed the treads down with concrete screws, then put wood filler on top of the screws, they did something similar in our wardrobes and after it's been stained and varnished you can't even spot where the screws were.
The problem in this part of the world is that the wood isn't usually left to dry for long enough so tends to bow. So best to make sure it's really screwed down. One of might mates had his whole kitchen made from the local teak and now two years on, not one of the doors close properly.