As you can see, very simple: just the +5V rail connected to pin 8 on the ATtiny and to the "left" pin on the TMP36, and the ground rail connected to pin 4 on the ATtiny and the "right" pin on the TMP36.
I measure the +5V rail at 5.116 using a Fluke 115 multimeter. Again, as soon as I hook up the ATtiny the output of the TMP36 drops to about 0.1 V. I hooked my oscilloscope up to the output just for kicks and it looks pretty normal too. There's a slight bit of noise when I plug in the ATtiny but it's just an extremely small ripple, the TMP36 should have no trouble with this setup.
Also FYI the ATtiny is just running the "blink" program just so I know it's not doing anything weird.