The Bilski decision is probably the one most relevant to this question. Unfortunately, the answer is that nobody really knows.
In the Bilski decision, the Court of Appeals for the Federal Circuit ruled that to qualify as a patentable method (rather than an unpatentable abstract idea), the invention had to involve either a specific machine, or a transformation of matter (e.g., mixing chemicals).
The Supreme court upheld the decision (i.e., rejected the patent in question as invalid), but specifically rejected the "machine or transformation" test that the CAFC had created. At the same time, they do fairly directly say that although the machine or transformation test cannot be used as the sole rule, that it is a useful guideline.
I'd take that to mean that if the invention does involve a specific machine or a transformation of matter, it almost certainly is patentable. Otherwise, it may still be patentable, but to do so you're going to have to convince the PTO (and quite possibly courts) that it really is not an abstract idea.
As an aside that's somewhat relevant here, earlier rulings have established that just "computer" probably doesn't qualify as a specific machine, but "digital computer" probably does.