Noah Smith shares some information on a point that some of you may already be aware of: modern business cycles aren't really "cycles". Booms don't correlate well with subsequent busts - instead, busts correlate with subsequent booms. Milton Friedman was one of the first to highlight this point - he called it the "plucking model". It's a simple insight, but I personally think it's one of the strongest arguments against Austrian business cycle theory. Indeed, Friedman pointed to Mises as an example of a theory that was rendered problematic by the plucking model. Garrison (and surely others) have arguments against this, but I've never found them all that convincing.
[UPDATE: So as I think about it, Friedman's plucking model point and Noah's trend-stationary point are probably a little different (although someone with more background in time series can expand on this point in the comments). Presumably, subsequent booms correlated with prior busts could be a genuine cycle theory which started with a bust. I think that sort of cycle theory is less natural to think about than a cycle theory that starts with a bust - which is why the plucking model is usually associated with the idea that the economy is growing and occasionally gets hit by shocks that dissipate over time (the line I drew above between Friedman and Noah's points). But I suppose that's not strictly necessary.]
Noah also gets into Minsky, who he talks about as having a more classical cycle theory. My understanding of Minsky (which is admittedly limited), though, is that it's probably more accurate to say that he had a cyclical theory of financial markets - which of course is a little different from having a cyclical business cycle theory. After all, financial crises often provide the shock that might cause a recession in trend-stationary GDP data. So the question for evaluating Minsky is to ask whether financial markets are trend-stationary or not. If they are, then a cycle theory may be less appropriate. I really don't know this literature, but my understanding is that's a disputed point - some early work said it was, but a lot of people are concerned about the robustness of those results.
If you're interested in these sorts of distinctions, you should check out some of Gene Callahan's posts on the subject. He is working on some kind of survey of cycle theories, focusing on those that offer actual cycles of the sort that Noah is describing (not just shocks that are called cycles, such as RBC). I'd just go to Gene's blog and search on "cycle theory".