Zener Diodes are extremely noisy. So much so that they are often used as noise generators in e.g. synthesizers. To make things worse, the amount of noise rises with the rated zener voltage. In most circuits this is not good but for T powered microphones this is much more critical since EVERY noise, hum etc of the power supply is directly added to the audio signal. Placing a large capacitor parallel to the zener helps only a little because the internal impedance of the zener is very low and to suppress the noise in you need a really big capacitor (think of 1000uF or so).
If you do not trust the circuit from my previour post and you really want the zener based circuit, at least replace the 12 volt zener by 2 6.2 volt zeners in series. That helps a little. Better is to use leds for the voltage srtabilisation. A blue or white LED had a forward voltage of approx 2,5 .. 3.0 volt, so with 4 or 5 in series you also have a reasonable stable 12 volt with much less noise.