Are you using the circuit in Fig 4 of the HIH6130 datasheet?
Yes, I'm using figure 4 (left side) in the datasheet.
If so are you also removing power from the pullup resistors and the LED?
1. If I use any pullups, it doesn't really matter, for testing I usually skip them (so I don't need t use a breadboard), for production they are there. In this case the result is the same.
2. There is no LED. In the datasheet the LED or whatever is there is to show the use of the alarm triggers. I'm currently not using them (that is pins 5 an 6 on the HIH).
3. I don't remove power from the pullups, the are wired directly to 3.3V on the arduino.
Also when you say "remove power" are you driving pin 10 LOW or making it an input?
I'm using: digitalWrite(X, LOW);