The Oyez Project is a multimedia archive of the Supreme Court’s goings on. Soon after the Court hears oral arguments, Oyez posts audio transcripts of the arguments for your listening pleasure.
Simply visit Oyez, click on the cases tab at the top of page, and it’ll bring up the list of cases from the most recent term. A side menu will allow you to bring up cases from past terms as well. Click on case titles to find a brief discussion of the case, audio transcripts of the oral arguments (for those cases between the 1960s and the present), a link to the full opinion, and pictures of the justices (you can switch to have them organized by vote, by seniority or by ideology).
One neat feature is that when you play the oral arguments, it not only plays the audio, but provides picture of who’s speaking so you can keep track of which justice is asking which questions. You’re also able to download the full transcript text.