Piggyback off Motherboard Power Switch?
For PC's made after the introduction of the ATX, the power switch is a logical input to the motherboard--the same kind of input you want to apply to the Teensy.   The motherboard, in turn, outputs a separate logic signal to turn on the power supply.

In my experience, the reset switch is as good as standard, as it dates back to the original IBM PC (which didn't have the switch, but did have a connection on the motherboard for one).  This is not so for the power switch.  I have seen schemes where it pulls a signal up and neither side is grounded.  Since you have found one side of your switch is connected to ground, that makes your connection easy.  Simply attach a voltmeter to the two pins, and observe the voltage present there.  Verify that this voltage drops to zero when pressing the switch.  Your Teensy will see this on its input.   It doesn't matter how they implemented this.  If it's a floating input they will have a pull up resistor already there to make it work.  Your input is just watching the resultant voltage level.

Note that the power supply is turned on when the motherboard drops the #PS_ON signal to ground.  This is the infamous "green wire" on the ATX connector.  If you bring this signal into a second input on your Teensy, you will now be able to tell if the button was pushed while the computer is already running (because it will show a logic 0).  

Also note that certain voltages are always present even when the computer is off (or in standby).  This is how the power switch manages to have power when there's no power.  This is also a heads up that you may find power in other unplanned places (like some USB connections).