Microsoft has confirmed to IGN that Xbox One will not come bundled with a headset, as per Xbox 360 tradition. Instead, Microsoft points to Kinect's "highly sensitive multi-array microphones designed to enable voice inputs and chat as a system-level capability, both in-game and with Skype and other experiences."
If you want a standard Microsoft Xbox One headset (which Microsoft calls "a must-have," ironically), you'll have to purchase it separately. Additionally, Turtle Beach headsets and Polk headphones will be available following the console's launch. For context, Sony doesn't default to packing a headset with PlayStation hardware, but the PlayStation 4 is bundling a mono earpiece and mic.
I'm starting to think now that Our Expensive headsets won't work with the Next Gen Consoles.