As anybody who has watched a courtroom drama knows, prior to testifying witnesses swear to tell "the truth, the whole truth, and nothing but the truth" while placing their right hand on the Bible. If the witness does not believe in God, then he can choose simply to raise his right hand and say an oath. Kind of quaint, really, but it's a nice tradition.
In North Carolina, a local Islamic group offered to donate some Qurans to the region's courts so that Muslims could swear on them instead of the Bible. And how did the judges respond? By rejecting the overture and proclaiming that an "oath on the Quran is not a lawful oath under our law."
I would say that such an action is blatantly unconstitutional, but North Carolina is in the 4th Circuit--the circuit perhaps most hostile to non-Christian religious freedom claims. I'm not sure this is any more outrageous than the anti-Wiccan policy the 4th Circuit upheld in Simpson v. Chesterfield County Board of Supervisors (exclusion of Wiccans and only Wiccans from giving a county's legislative prayer is permissible because a "divine
appeal [must] be wide-ranging, tying its legitimacy to common religious ground" in order to be constitutional under the first amendment), so this policy may yet survive. It need not even be said how this will affect our reputation with Muslims who already are sympathetic to portrayals of America as an anti-Muslim nation.