Same here - I didn't think about that aspect either. And by no means am I an expert in construction (just the average DIYer around the house), but am pretty sure you're going to need some concrete block or a poured concrete structure.
Well, concrete would be ideal but this will pushup the costs. When you pour concrete, you'll need to do proper formwork, and that on it's own is a structural consideration due to the wet concrete pressure. Then there's the reinforcement and concrete cost itself.
It could be done by wood, but just not with a few odd bit and pieces here and there.
If smartly designed, you could have a desent and safe structure.
Another thing, the structure (wooden or something else) will need to be properly founded...it'll need a foundation of some sort to anchor it...otherwise the main elements would kick out at the bottom and guess what'll happen then.
I'm just a structural engineer, but these are the things I'd look at when considering to build a cool underground place.
