The longest-outstanding case is Bilski v. Kappos, which involves the patentability of "business methods." Bilski was argued in November. The only Justice who has not issued a majority opinion from that sitting is Justice Stevens, which makes him the very likely author. Justice Stevens tends to take a narrow view of patent rights. He notably joined Justice Breyer's opinion in Laboratory Corp. v. Metabolite in 2006 arguing for a narrow interpretation of process patent rights, which is a similar issue.
At oral argument in Bilski, Justice Stevens was very engaged. He asked counsel for the patentee the following telling question: "But is it correct that there's none - none of our cases has ever approved a rule such as you advocate?" Justice Stevens also was seemingly doubtful that the involvement of a machine could render a process patentable, and furthermore that software could be patentable, which suggests a very narrow interpretation of business method patents and that the ruling could spell significant trouble for software patents.
I ultimately predict that the Bilski majority opinion will be authored by Justice Stevens and that the decision will be very significant in its narrowing of the scope of method patents. I expect that the delay in resolving the case will have arisen not from disagreement over whether this particular invention is patentable - I think the Court will unanimously hold that it isn't - but over the scope of the rule.