I've got it working in TeamSpeak. It doesn't sound nearly as cool through that as it does normally, but it's still pretty nifty.
The EASY way to do it is to use two devices connected to TeamSpeak. One you talk through, the other one plays Syrinscape. You'll have to set the Syrinscape device's default Recording Device under Sound Settings to Stereo Mix. This takes what comes out of your speakers and dumps it into your microphone. If your sound card doesn't support Stereo Mix, try using a program like Virtual Audio Cable or VB Cable. Check out the link in the hard section for help setting that up.
The HARD way is if you only have one device that can use TeamSpeak.
1. Download a program that adds a virtual audio channel. Virtual Audio Cable is supposed to be good; I use VB Cable since it's free and works for this.
2. Open the properties of your TeamSpeak shortcut and add "-nosingleinstance" after the Target path (no quotations).
3. Follow these directions starting at Step 2 to set up a profile for Syrinscape on TeamSpeak. Syrinscape doesn't have an option for Step 5, so read on.
These steps need to be performed each time you use the program.
4. Right-click on your speaker icon in the System Tray and select Playback Devices. Select Line 1 or CABLE Output or whatever the program you used added and make it the default device.
5. Launch Syrinscape. Make any audio settings changes you want to. I set it to Mono since TeamSpeak is mono output.
6. Return your default playback device to what is normally is. You will now not be able to hear Syrinscape except through TeamSpeak.
7. Launch TeamSpeak twice and use your bookmarks from the tutorial to log in as yourself and as Syrinscape.
8. Everyone should right click on the Syrinscape profile and adjust the volume to a comfortable level. -15dB or so is usually good.
That should do it.