They heard cases without him, using the same procedures for cases when a justice recuses him/herself. For instance, Girouard v. United States (1946) was argued and decided (5-3) without Jackson's participation since he missed the entire October 1945 term. I don't have a list of cases from that term, but Supreme Court deadlocks at 4-4 mean the decision of the next lowest court is upheld.

From the Robert H. Jackson Center:

"Jackson’s acceptance of President Truman’s request in 1945 to become the U.S. chief prosecutor at the Nuremberg Nazi War Crimes Trials, and his subsequent absence from the Court for an entire term, compounded his difficulties with his colleagues – not to mention the business of the Court, which had to operate with eight justices and the attendant all-too-present danger of 4:4 tie votes of which were a score."