I think the simple answer is: put your butt/hips into the kick. The hips generate a lot of power, and this has to be made use of while kicking as well as punching.
For example, in front kick, thrusting your hips forward gives the kick power. Imagine that at the instant of throwing the kick, someone slaps you forward hard in the small of your back. That is how our instructor gets us to kick with more 'snap' (see also the comments section for a reader's suggestion about how to think of the hip action). In roundhouse, during the kick, your hips should be moving 'into' the kick. To do this, bend the balance knee so that your hips are free to move forward in the direction of the kick. In back kick, try to keep your hips low while kicking and use your hips to thrust back, while you hold the kick for a vibration.
I want to say that I'm a beginner and learning these things for the first time, so I don't want it to sound like I'm an authority of any kind. I wrote this post because I felt it might be helpful for me (and hopefully other beginners) to put what I am learning about technique into words as precisely as I can. So if readers have corrections, suggestions or thoughts, do comment!