|The saying: "Swimming is all about technique"
is not entirely true. Fast swimming is about good technique, but also how much power you can develop from the appropriate muscles.
Great swimmers have a balance of both technique and power.
I've watched swimmers with beautiful form and no upper body strength who look graceful moving through the water, but don't have the power and turnover rate to go fast. I know swimmers with a football, wrestling, or weight training background who make me look like a wimp in the weight room, but have no feel for the water. These same athletes also often lack flexibility through the shoulders so their stroke is short and choppy.
So my recommendations to swimmers who look to gain speed are based on where they need work. Sometimes weights and cord work is needed, while sometimes it's all about form; all swimmers need a minimum amount of time in the pool to master it.
The notion that one swim coaching group uses in their marketing propaganda that swimming should be "effortless" is misguiding. If you swim for general fitness it should be a relaxing activity—but if you swim competitively the effort is similar to cycling and running; it hurts to go fast!
For swimming the muscles that should be doing the most work are your lats. Fortunately these are some of the most powerful muscles in your upper body and rarely if ever get strained. Your pectorals and triceps should be doing some of the work as well. Core muscles should also be engaged and are crucial to hold optimal body position. Elite swimmers who already have sufficient upper body strength typically spend more time doing core work than weights.
How many unassisted chin-ups and dips can you do? If the answer is "none," you should do some work there as strength will be your speed limiter even if swim technique is optimal.
You don't need a great kick to swim well, but it helps. If you struggle to kick a lap you should dedicate some pool time to improving. Your kick helps to hold the legs up as well as directional stability. In a wetsuit you can get away with no kick, but without it you need some kick to stay horizontal. Elite swimmers do have a powerful kick and it contributes significantly to their speed, though the time it takes in the pool to develop this level of power is not worth it for most of us.