For me, technique needs to be equated to the sound produced. It is our connection to the instrument without which we really cannot play well. If you think about it, the art and science of technique for any instrument is the study of how we physically control the properties of sound -- pitch, volume, duration, and timbre -- and learning what we can and can't control of those properties.
Technical study should not just use a bunch of mindless finger exercises that can be practiced while reading a book or watching TV (yes, that has been advocated in the history of piano instruction). The student must listen as they practice. Therefore the teacher needs to give good sound models that incorporate the technique in question. This is done both at the lesson and can be done through recordings of exercises and pieces that incorporate the techniques being taught.
At the early levels, particularly for children, technique is very much a rote experience. It needs to be modeled and copied. Where possible, it should be related to motions that they already do naturally. This is particularly important in the earliest lessons so that their success is insured and they will then be open to more advanced techniques that do not feel as natural.
For me the first technique to learn and reinforce is proper seating and posture. The height of the bench must be properly adjusted so that their hands and arms are properly placed, and, if their feet do not reach the floor, they must be given something to rest on. This is easily done with old city phone books covered in contact paper or an old set of encyclopedias or coffee table books that you can pick-up for cheap. This can also help them focus on proper seating because as they grow the will discover that they can use fewer books and are on their way to sitting right on the bench like older people do. They learn that if they don't sit properly, they won't be as aware of their growth. So, at the lessons, have the student make the appropriate adjustments to the bench.
In technique in the earliest level, I prefer to use the R.H. at the "C" above middle "C" and the L.H. the "C" below middle "C". When properly seated, this is more natural. The arms are straight forward, not bent in toward the middle of the body that is required when starting with middle "C". Most methods use the middle "C" simply because that is the easiest place to start with notation.
The next technique that I like to teach is simply playing to the "bottom of the key" -- in otherwords making sure that the key goes all of the way down regardless of how loud or soft the sound is to be. They can experiment with this by playing the key from a little above the key, which will make louder sounds, or starting right on the key, which will make softer sounds. Again, they must be listening as they feel for the key bottom.
The first playing techniques that I use are actually from the earliest level of Suzuki. In this, there are 4 variations of "Twinkle, Twinkle Little Star" that work from large muscles to small muscles. The first variation incorporates the rhythm of 4 - 16th notes and 2 - 8th notes on each note of the song (you can actually start by using a 5 finger pattern) -- often called "Mississippi Hot Dog". The motion incorporates the large forearm muscles and is natural in young children (think about a baby getting excited and shaking their forearms). It is very similar to the motion used to knock on or door or a table top and it practically insures that they will play the key all of the way down. Since the motion is fairly natural, you can have the student listen carefully to the sound to make sure that each note is the same volume, whatever volume it is (though not too soft). Since the release of each note incorporates a natural rebound, they don't have to worry about any special release motions. It also is almost impossible for them to play the rhythm wrong. It also helps to relieve a build up of tension from the beginning.
As they are working on this variation, you can start working on training the smaller muscles that control the finger to create curved fingers (sort of like a letter "C"). The only explanation for "why" need be that when the fingers are flat, they are different lengths. When we shape our hands into a "C" our fingers appear to be the same length because they play the keys at the same place.
The second variation uses 8th quarter 8th -- "Ice Cream Cone" and begins to develop the idea of a release on the long note by guiding the finger off the key by lifting the wrist. Here, they are listening for evenness of tone as well as a long sound on the "Cream".
The third variation uses 8th 2- 16th 8th 2-16th -- "Run Mommy, Run Daddy". This is similar to the first except that it tends to use more wrist motion on the 16ths. Again, listen for evenness of sound.
The fourth variation is the actual song rhythm. This variation begins the concept of legato and incorporates more finger motion. Even the first of the repeated notes is held as long as possible to create the illusion of legato and the student listens for smooth, connected sounds as well as evenness of tone. If during the first three variations, you are working regularly on shaping the hand, their fingers will be nicely and naturally curved for this variation.
These variations have helped to develop some of the most basic motions used on the piano -- forearm, wrist, wrist release, finger legato as well as expansion of the hand used from thumb to 4th finger (C - G) as well as contraction in the tune of "Twinkle". Additional techniques are built from here as well as exercises for finger dexterity. In addition, many of the problems associated with the interdependence of fingers 4 and 5 are avoided or minimized at this point. I have found that that interdependence is not as big of problem and exercises and music requiring greater independence of those fingers are more accessible.
I hope this gives you some ideas.