Human voice is generally around 100-900Hz, so you don't need super fast stuff to get the fundamental pitch, but there are higher overtones in the sound as well as in articulations (start and stops of syllables or other percussive noises). If you're familiar with musical pitches, the A in the center of the treble clef staff is 440Hz, so the one above the staff is 880Hz (and is somewhat high for a Soprano singer) and the A at the bottom of the bass clef staff is 110Hz (lowish for a Baritone/bass).
It sounds like what you're looking for is a tuner, but perhaps without the note letter display and just a frequency readout. You could take the input from a mic, amplify it, and then use a comparator and a digital timing circuit to time the period of the amplified wave, but it takes a fair amount of finesse to get that to be consistently the sound you're expecting. Generally you'd want some sort of automated gain control (or automatic attenuation, depending on your configuration), and you may have to play around with the comparator trigger point to get a reliable detection of the tone you want, but it could still be tricky to get in the presence of noise or with certain timbres of sounds measured.
One trick could be actually to dramatically distort the audio waveform by amplifying it to the point of clipping, then using your comparator at the center. That makes the fundamental a little easier to pick out by pushing lower harmonic content into the clipped regions of the signal, but then if the signal you're interested in is slightly lower amplitude than higher frequency content, only the higher frequency reads out, even if the lower one sounds louder (human ear frequency response is far from linear).
Another interesting option could be a variation on a strobescopic tuner. basically, you spin a physical disk with black and white markings on it at a fixed speed (controlled by an oscillator and a stepper motor driver), then flash a light against it using the amplified audio input as your light trigger. The resulting pattern makes the lines on the disk appear to stop moving when the pitch going into the light is the same as the speed of the disk (well, speed of the disk and number of lines per revolution). You'd have to manually tune it around to find your frequency of interest, but with extra bands on the disk, you can see several octaves at once.
If you went the DSP route, you could get a pretty good idea of the frequency and amplitude of the tone with an FFT. You run an FFT on the band of interest and search for the peak, then read out the results.