Will any headset without a microphone work with the Xbox one headset adapter?

Almost all excetp the oringial XB360 headset and few surround headsets.

What adapter did you try without success?

Be aware that there are two standards for 3.5 mm headset jacks called OMTP and CTIA. With both standards, the microphone and ground pin connectors are reversed.

Which means, if your Bose headset has an OMTP standard jack, the headset is sending microphone signals to the controller's ground, while the controller is "listening" for the microphone on the headset's ground pin.

If the headset actually works on your smartphone for example, then simply get an OMTP to CTIA adapter cable. Since only two wires get crossed over, these adapters should be dirt cheap. If the store is asking for a higher price than a coffee-to-go for such
an adapter, you are being ripped off. Also, since crossing the ground and microphone pin works both ways, you can also get a CTIA to OMTP adapter, as both are the same thing.

Update: Regarding issue #2 (Headaet adapter issues while connecting Xbox One controller to Windows 10 PC for streaming)

I discovered that:

1) The headset adapter is just treated like an additional headphone and microphone under the Windows audio settings.

2) To select which microphone to use (e.g. PC mic or microphone attached to Headset Adapter), set the desired microphone as your default device by right clicking volume/speaker icon at the bottom right of the taskbar and click "Recording devices."

3) Odd behavior or malfunctions (e.g. headet microphone not working when streaming to Wkndows 10) may occur when there are too many USB plugged into the computer, causing insufficient USB controller resources.

When I plugged the Xbox One controller directly into my Surface Pro 2, the microphone automatically worked fine. When I docked my Surface Pro 2 into its docking station and plugged the Xbox One controller into the docking station's USB port, my headset adapter/Stereo
Headset's microphone was recognized, but the decibel meter would not detect any sound when spoken into. Occasionally, I would receive a "insufficient USB controller resources" error. This could also be an issue related to connecting the Xbox One controller
to a USB hub.

Regarding issue #1 (headset adapter/stereo headset's microphone not working on Xbox One after disconnecting from streaming), I will see if I experience it again to isolate the issue.

Has the microphone worked on the Xbox One before?

If you are trying the headset and microphone for the first time, make sure that your headset is using the CTIA standard and not the OMTP standard for it's plug. For more information about these plugs see this website: https://support.xbox.com/xbox-one/accessories/connect-compatible-headset

If your headset has an OMTP instead of CTIA plug, the mic and ground connectors on the plug are switched.

You can then simply buy a "OMTP to CTIA" adapter (since the adapter only crosses two connectors, they work both ways).

Just tested this and it will work.

Xbox One connected to a monitor using an HDMI Cable. Set the Display settings to DVI. Meaning no audio is being transferred over the cable. Attached Headset Audio adapter to an updated xbox one controller and plugged in the Xbox One Stereo headset to the
adapter, got Stereo sound on the headset without any issues.