Speed has a great influence on all techniques in karate. Without speed, most techniques would be of little value. Is speed everything? No. You must first learn a technique well, which can be best achieved at a slower speed, then add speed to the technique once learned. Thus, speed is not the end-all in karate, but maximizing speed is critical during kihon (basic) training.
Maximizing speed comes from maximizing the speed in your legs, torso, and arms. That is, each part of the body that moves in a technique needs to move at its fastest speed. Let’s consider a step punch. The legs move fastest by contracting the adductors (inside thigh muscles) as quickly as possible and then exploding off of the pivot leg by thrusting the heel into the ground. To maximize the punching speed, the withdrawing hand is withdrawn as fast as possible. Withdraw faster = punch faster.
Speed is also dependent on muscle control. If unnecessary muscles are being tensed the speed of the technique will be slower. For example, during a punch the biceps are relaxed while the triceps are contracted. Contracting the biceps also will contribute to a slower punching motion.
Shoulder muscles are usually relaxed during techniques. If you become aware of rising shoulders, consciously relax and drop your shoulders.
So, learn to be aware of which muscles you’re tensing at what times. Concentrate on relaxing those muscle groups that are not contributing to the power or speed of a technique.