If you head down to D.C. Superior Court on any given weekday, you can walk into a courtroom, watch a trial and take as many notes as you want. But if you pull out your smart phone and start tweeting away, a U.S. Marshal will likely approach and remind you that those 140-character updates are verboten, as is any type of electronic recording.

That policy extends to many courts around the country, but as the AP reports today, civil liberties advocates are pushing for judges to lighten up on courtroom tweeting:

“We’re troubled by this ban,” said Ed Yohnka, Chicago spokesman for the American Civil Liberties Union. Tweeting and social media are “merely the 21st century version of what reporters have always done _ gather information and disseminate it.”

Judges, he said, should embrace Twitter as a way to shed light on the judicial process, which, for many Americans, remains shrouded in mysterious ritual.

Some judges, though, aren’t enamored with the idea—a courtroom full of people staring at their phones could disturb the proceedings.

Last November, D.C. Superior Court Chief Judge Lee Satterfield issued an administrative order specifying that while computers and cell phones were allowed in the courthouse, they could not be used in courtrooms. While journalists can petition for exemptions, they still wouldn’t be able to tweet—the order specifies that “Transmissions of any type shall not be sent or received.” Still, even here there might be some flexibility. According to a media guide published by the D.C. courts, individual judges can allow live-blogging of proceedings, which would likely include live-tweeting. (D.C. Courts spokeswoman Leah Gurowitz confirmed that judges have the final say.)

The argument over Twitter in the courtroom got national attention when a crafty conservative activist managed to tweet from the U.S. Supreme Court’s arguments on President Obama’s health care law. He was eventually asked to stop; the court doesn’t even allow cameras, after all, and usually waits until the end of the week to release transcripts of arguments. (It made an exception for the health care law.)

Obviously, jurors should completely refrain from tweeting during trials or grand juries. That doesn’t mean you can’t tweet away while waiting to be called in for jury questioning, though—the courthouse does have great WiFi, after all, and the movies in the juror lounge usually suck. Still, should the media and court watchers be able to tweet from the courtroom?