Ya I took karate for many years also, and the "hinge" is a good way to describe it.
Basicly just throw the kick like you're swinging a bat (your leg) around and through the target. Don't chamber or cock the leg bending it at the knee. Create tension on your hips by using a combination of turning your upperbody and stepping out at an angle with your supporting leg (or for those of you that favor the foot pivot use that pivot to create the hip tension) and when the tension is max, release it swinging your leg horizontally through the target. Practice throwing the kick without a bag a few times and do a 360 degree turn throwing it to get the right mechanics. Now throw it into the bag with the same intention that you would optimally go through the oponent and turn around completely.
And viola, you should have the makings of a good thai roundhouse.
Something to note is that you get more power stepping to the outside of your target as you throw the kick. Being straight on with the target shortens your kicking distance, and the longer your kick travels the harder it hits.
Be wary of instruction to kick at upward or downward angles into your target. Firstly, this can have you injuring your foot kicking up into an elbow. Secondly, the best way to impact with a thai kick is in such a way that your kick impacts straight in. Maximum tissue damage and pain is a result of compressing "meat" between bone most efficiently. This is done by kicking into the leg (for example), as perpendicular to the way the bone is as possible.
Some thai instructors try to promote a downward "chopping" leg kick for example.
