My 2000 Jetta TDI blows cold air while driving. No fun this winter! I would expect a bad thermostat; but I understand if the thermostat is bad a warning light would be lit on the dash. Is that the case?

No, this is not your case. If the thermostat is bad any warning light would be lit on the dash.
First time you might check the thermostat operating.
Heed of antifreeze coolant temperature!!!
The thermostat location for 2000 VW Jetta 1.9 TDI, is on back of the alternator at the bottom side of the bracket.
IMPORTANT: It is NOT necessary to replace the alternator if you want to change the thermostat !!!
Thermostat checking:
1. Cold engine. Start up the engine and watch of the water temp indicator. After first value 60 Celsius Degree you still warm up the engine until the lifter (pointer) reach 80 C deg. (second mark after 60 C deg.). At this time you let engine running at idle speed and check with your hand the thermostat plastic housing and hose temperature in bottom left side of the engine. Heed of alternator driving belt and ventilator fan !!!
2. If the thermostat is good the hose must be cold until the water temp indicator reach almost 90 C deg.
3. The thermostat opening temperature is 87 C deg.
4. If the hose is already warm the thermostat is out of order.

Thermostat location, Part number: 044121113, in german - "Kuehlregle" (Price around $25):


Important note: Always replace the thermostat round seal, position 7 in the above picture.

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