Learn Farsi: یک حرکت پیوسته در راستای ریتم لازم است. - A continuous movement in accordance with the rhythm is necessary.

Yes! It literally means along or in the direction of, coming from the noun râstâ (direction or alignment). While it is used abstractly here to mean in accordance with the rhythm, you will also frequently see it used for physical spaces, such as moving along a specific path or axis. Remember that it always needs an ezafe to connect to the following noun, which is why it sounds like râstâ-ye.

Why use peyvasteh for continuous instead of a word like modâvem?

Both are fine, but peyvasteh comes from the verb peyvastan (to join or connect). It gives a specific nuance of being seamlessly linked together without any gaps or breaks. For a physical movement matched to a rhythm, it describes a smooth, unbroken flow perfectly.

Could I rephrase this using bâyad instead of lâzem ast?

You could, but it requires a totally different sentence structure. The word bâyad (must) needs a conjugated verb in the subjunctive mood, so you would have to introduce an action and a doer, for example saying: we must have a continuous movement. Using lâzem ast (is necessary) keeps the sentence focused purely on the movement itself as an objective fact.

Just to double-check, where are the unseen ezafe links in this sentence?

There are two! The first links the noun to its describing adjective: harakat-e peyvasteh (continuous movement). The second connects our complex preposition to its target: dar râstâ-ye ritm (in accordance with the rhythm). Because ends in a long vowel, the ezafe takes the sound.