I have the following circuit which is intended to turn on LEDs depending on the voltage from the right hand side of R3 (eventually the LEDs will be logic inputs). For some reason the Zener diodes Z4, Z5 and Z6 conduct well below their threshold voltage (shown is the 4.7V Zener conducting at 3.32V). I've tried different Zeners with the same result. How should the Zeners be connected to only turn on the LEDs when at or over the respective threshold voltages?
Zener diodes do not have a sharp transition. Lower voltage ones use the Zener effect, higher voltage ones use the avalanche effect. Zeners rated under 6 V tend to have a rather soft 'knee' in their characteric curves.
If you need something with a sharp cutoff you should look into voltage references and comparators. For your case you could use something similar to this circuit replacing the 9 V battery with a voltage reference or regulated voltage. The resistive divider values can be adjusted to give the voltages you need. If you want to only have one output on at a time (rather than a progression) you can use window comparators.
Image by Fred the Oyster - File:I-V curve for a Zener Diode.jpg, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16129785
Never connect a voltmeter in series with anything, ever. A perfect voltmeter has infinite resistance, and will pass no current at all:
On the left, which is bad, the voltmeter tells you nothing about about how much current is flowing, but one may infer that no current is flowing, because the voltmeter has super-high resistance. The placement of the voltmeter in series with D1 ensures that D1 never illuminates, since current must flow via D1 for that to happen.
On the right is the correct way to employ a voltmeter. It is measuring the potential difference between C and D, otherwise referred to as the "voltage across D2". It doesn't explicitly tell you anything about the current through D2, but at least it isn't impeding that current in any way.
In the left circuit, the reading of 12V on VM1 has nothing to do with the state of the LED, except in the following sense: voltmeter VM1 has 12V across it, which means that of the total 12V available from battery BAT1, VM1 is "taking up" all of it, leaving no voltage across R1 or D1. This is in accordance with Kirchoff's Voltage Law. The LED is emphatically OFF.
The placement of VM2, VM3 and VM4 in your own circuit is incorrect, ensuring that no current passes via the LEDs, and your statement that the zener diodes are conducting is false. Those LEDs are all off, regardless of the potential shown on VM5. The readings on VM2, VM3 and VM4 are not an indication that the LEDs are lit, only an indication of voltage that the LEDs do not have across them!