Many libertarians latch on to austrian economics because they already agree with the conclusions austrian economists come to, and it's easy for lay people to understand. Conversely, it takes quite a bit more effort to understand a neoclassical dynamic partial equilibrium model.
As the other poster mentioned, the Austrians explicitly reject statistical or modeling methods, and usually argue with words, dressing them up as if they are "pure logic". The problem with arguing with words is that it is far easier to decide your conclusion and then grope towards a justification in words than it is to start with some assumptions and figure out what the conclusion should be.
When I read the Austrians, I never get the same feeling of dispassionate honest investigation that I do when I read any other science (or even soft "science", like econ). I just feel like they've always already decided the answer before they started working on the problem, and that rubs me seriously wrong.
Not that I think most other economists are much better, to be fair. And the Austrians may be right in some of their claims. But I get the distinct feeling that they are so committed to their conclusions that even if they found evidence against them they would pay it no attention.