Specifically, I have a PowerColor HD 6850 Passive card. How do I verify that it's not overheating?

Furmark

Stress test that monitors temps.

Use MSI Afterburner.

2nded, I use this myself.

GPUz will display all temps, clocks, load, etc. Careful running furmark on fanless though, stop it if you passing 90C.

Furmark will do the stresstesting, display temps (also graph them), display GPU voltage, and also display fanspeed as an RPM and a percentage.

In short, one app that will do everything you need for measuring.

If you need to actually modify settings - that's when you want to progress to MSI afterburner (I just use Catalyst Control Centre, as I don't mod voltages)

I'm not quite experienced with this, but I've used GPU-z, Furmark and CCC in the past. I've noticed that CCC and GPU-z show similar temps, but differ from Furmark with about 4-5 degrees Celsius.

MSI Afterburner does way more than this. It is an essential piece of FREE software for OC'ing, voltmodding and silent computing (fan management). It totally rocks together with a Logitech G15 keyboard with display. Allows you to see all the info on the display while you game of stresstest. Lovely.