Oh dear, this seems to be one of those situations, it clearly should be working but it just isn't.
Unless my eyesight has gone too far then it looks as if you have the pinout correct according to the datasheet. I don't know where you sourced the LM334s from, but if it is from a Chinese ebay seller then it is quite possible that all 10 could all be faulty - or rather re-marked something else.
The other thing that comes to mind is that you are using a breadboard with various thickness wires. I never rule out contact faults. I noticed when you tested the setup with the 10k resistor, you touched the resistor to the red crock clip rather than the V+ device pin, so you haven't eliminated a disconnect there. Also, if either end of the resistor is disconnected, then the LM334 will drop to sub uA quiescent leakage current which wouldn't register on that meter range. For such a simple test I would normally just spread the legs of the device slightly and tack solder the resistor between the R and V- pins, everything is clearly visibly and solidly connected then.
I was momentarily confused by the schematic as you have put -Ve at the top, but it is logically correct.
Sorry, I can't give you a definitive answer, but it looks right to me, so it must either be a device issue or dodgy connection.