According to this 2002 Media Institute Report Judge Edith Clement (who many believe will be Bush's nominee to replace O'Connor on the Court) granted a motion to anonymize a jury in a corruption trial which stated “Any attempts by the media or others to interfere with [juror anonymity] will not be tolerated” and ordered the media “not to circumvent this Court’s ruling preserving the jury’s anonymity.” When pressed by news organizations on what was meant, she clarified that the order covered reporting on the identity of jurors no matter how the information was obtained. The U.S. Court of Appeals for the Fifth Circuit upheld the judge’s order keeping jurors anonymous, but also said that stopping journalists from reporting the names if they obtained them independently was unconstitutional. United States v. Brown, 250 F.3d 907 (5th Cir. 2001).
If Judge Clement was so easily persuaded to impose "an unconstitutional prior restraint on publication or an unlawful restriction on newsgathering activity" in that case, I'm not optimistic about her willingness to protect newsgatherers' First Amendment privileges in other contexts when the issue comes before the Court. With the media under assault from the judicial branch, I assume this will come up in hearings (if she is nominated). I'll be interested to hear what she has to say.
UPDATE: Orin Kerr says Bush will announce the nomination tonight at 9PM EST.