The problem appears to afflict a select group of devices that properly handled Xbox 360 audio prior to the New Xbox Experience, such as the BenQ G2400W monitor. It is believed that the issue stems from an updated HDCP handshake--the protocol used to establish an HDMI connection--which the older devices are not compatible with.
Now, Microsoft has acknowledged the issue in a statement to Shacknews, apologizing for the "inconvenience" and noting that a fix is in the works. Said Microsoft:
While the vast majority of our users worldwide are enjoying the New Xbox Experience with no problems, we are aware that a handful of Xbox LIVE users are experiencing audio issues, and are diligently monitoring this issue and working towards a solution. We appreciate our members' enthusiasm and patience, and apologize for any inconveniences.
A temporary fix was also discovered--remove the hard drive, boot the system, then insert the hard drive after the dashboard comes up--but the workaround does not function for everyone, and must be performed each time the system is turned on.
Thanks to Shacker Vergs for the heads up.