Lots of great answers. The folklore/historical reasons are enlightening and make sense. However, I see a couple points that keep coming up, but doesn't really seem to hold much water:
1) Dwarves are basically vikings.
Am I the only one who associates vikings with swords more than axes? Or at least swords as much as axes. Does the phrase
"Viking sword" ring a bell to anyone else? What about
Hagar the Horrible?
2) Dwarves are craftsman/woodcutters, therefore they would use their tools in battle.
I assume most everybody has held and used a woodcutting axe, a maul, an adze, or some similar tool. But how many of you have actually held a war axe? A friend of mine has one (1-handed), and when I first held it, I was extremely surprised at how differently balanced it was, compared to a hatchet or axe of similar size.
A woodcutting axe is extremely top-heavy, since all you need to do is swing it straight in one direction. A war axe is balanced--perhaps more so than some short swords--letting you easily change directions in mid-swing. Trying to use a tool-axe in combat would be foolish, since missing on your first swing would likely spell your death.
So I would assume dwarven armies, like all other armies, would use weapons specifically made for combat, and combat alone. Which brings me back to my first point, which is that a small thrusting weapon would make more sense in a cramped cave than, say, a dwarved battle axe.
3) Dwarves need lots of wood to support their mines.
You know, I honestly never even thought of dwarven tunnels as having wooden supports. I mean, tunnel systems large enough to have cities in would be impossible to make using medieval mining techniques. I always assumed dwarves were such masters of the earth, that they would make tunnels without the use of supports, much like a natural cave. Such construction is a fantasy of course, but this is dnd, after all. For fuel, I always assumed they would use coal, oil, or some other fossil fuel found underground. The idea of carrying logs hundreds of miles underground to the farthest reaches of their tunnels does seem intriguing though...