Electric guitar pick-ups

AFAIK, the pick-ups are little microphones which “pick up” the sound of the strummed string and delivers the music to an amp.

This has revolutionized Rock since the technology has been implemented in 6-strings, 12-strings, and basses.

Has pick-up technology been ever implemented in other string instruments, like banjos, mandolins, sitars?

How about a piano?

Pickups have been placed in banjos and mandolins. I’m not sure about sitars.

The sound of a piano doesn’t just come from the strings. What you really want is the sound coming out of the entire structure. Typically a piano uses 2 microphones to try and capture the sound. If you are going to ruin the sound of a piano by shoving it through a pickup you might as well just use an electric piano. The digital ones are actually pretty decent these days.

Typical electric guitar (or bass, etc. ) pickups aren’t really microphones at all. They have a permanent magnet and a coil of fine wire and the vibration of the metal strings induce a current in the pickup. Thus, they must have metal strings in order to work, and the sound that they produce is not the same as the acoustically transmitted sound you hear directly. I would think a piano with magnetic pickups wouldn’t sound much like a piano (as engineer_comp_geek pointed out)!
There are also piezo-electric pickups, which are much more like a microphone than the standard magnetic ones. They can be installed on any intrument that vibrates. Often, these are used for acoustic guitars and other instruments not normally fitted with magnetic pickups.
Pickups of one type or another have been used on practically every kind of string instrument out there. Rickenbacker makes an electric, solid-body mandolin ( http://www.rickenbacker.com/us/5002v58.htm ) Other electric mandolins have been made as well. Fender, among others, makes an electric violin ( http://www.fender.com/gear/gear.php?partno=0950010 ). There have also been some purpose-built electric sitars ( http://www.stevesmusiccenter.com/StarsSitar.html ), but they don’t much resemble a standard sitar physically. Nevertheless, I don’t see any reason why at least piezo-electric pickups couldn’t be fitted succesfully to a sitar.

Electric guitar pickups: electro-magnetic (as Enigma says)
Acoustic guitar pickups: piezo-electric (crystal that produces electricity based on vibrations)–some of 'em are pretty darn good
Other instruments: A company called Barkely used to make pickups for darn near every instrument ever made–dunno if they still exist

All those solid-body mandolins, violins, etc. use piezos built into the bridge.

The Barkley flute pickup was a microphone built into a cork that replaced your head joint cork. It was pretty cool.

There were electromagnetic pickups for piano once upon a time (by Helpinstil?), but now that sampled pianos are a dime a dozen, almost nobody takes an actual acoustic piano into a club or on the road anymore.