Looking at the data sheet for a typical zener (eg BZX79C6v8
http://www.supplyframe.com/partsearchservlet/partnerWormhole.action?id=2345676&partnerName=DSA)
The dynamic impedance (Zz) is 30 to 80 ohm at 1mA so the attenuation of noise / rubbish on the 12v supply will be around 40dB (
~Zz/R1) so unless the 12v supply is very clean you will have a problem.
Similarly noise on the output of the 5v regulator will be amplified by the full gain of the circuit (10,000).
To avoid these problems...
First decouple the supply to the zener very heavily by splitting R1 and adding a large capacitor from the junction of the two parts to ground.
Second loose the 5v supply completely.
To do this DC couple the circuit by removing R2 & R3, shorting C4 and C5 and replacing the 5v supply to the lower end of R6 & R7 with a capacitor from each to ground (using 100nF will give a 16Hz LF roll off as before).
This will give the amplifiers unity DC gain so their outputs will sit a the zener voltage (rather than 5v + 100 times the offset voltage)
Jim