So I decided to just press on in a different direction. I changed the camera angle to behind the player, and slightly to the right of him (trying to emulate what I've seen of Gears of War ;)). I also brought out an old 3DS Max exporter that I wrote, exported some geometry, and wrote some code to read that into the engine. Finally, I read through this excellent article and managed to implement collision detection for the character (which seems to work 99% of the time...).
So you can now run around in a little (un-lit) level, without going through walls. The rest of the game is still in there at the moment (although everything that was meant to be viewed top-down looks totally wrong when viewed from this angle, like the blood effects etc.).
I'm not really sure what I want to do with what I've got now; I'm pretty confident that I don't want to finish the game as it was (a clone of Crimsonland), but I'm still quite keen on having ultra-simple gameplay.
My current thoughts: keep it outdoors-based, where the player shoots oncoming hordes of enemies, but have little "towns" consisting of indoor areas, which the player can go back to occasionally. In the town(s) he can spend cash (buy items, etc.) that has been earned off killing monsters. So... very much like Diablo, except 3D, and without any of the dungeon/quest-type stuff.
Anyway, all of this stuff I'm adding to the engine is relatively reusable, so even if I change my mind about the above paragraph within the next few days, I won't have been wasting my time ;P. Next step will be to generate some lightmaps for this geometry.
Finally, a screenshot of some geometry with a single texture on it... the ground plane that I was using before is still sticking through there.