From my own personal experiences, I manage to make a rhythmic style of writing by making my sentence lengths dynamic, from long to short to long again, or some medium in-between. This creates a smooth flow, and still follows the basic rules of English — excluding run-on sentences. They should be generally avoided, but I've seen even that as a style before. Dynamic sentence lengths create a flowing rhythm, and could be tweaked to even create different moods, like growing anxiety with a long and heavy sentence, and a sudden surprise with one little clip. As it says on this site:
Slow, descriptive or explanatory.
Can create a sense of relaxation, flow, or time dragging.
Using long sentences can create a fluent style and rhythm.
Good for action, and dramatic lines. For example, 'a shot rang out.'
Short sentences can create a punchy choppy rhythm.
An even more perfect example is stated by Gary Provost:
This sentence has five words. Here are five more words. Five-word sentences are fine. But several together become monotonous. Listen to what is happening. The writing is getting boring. The sound of it drones. It’s like a stuck record. The ear demands some variety.
Now listen. I vary the sentence length, and I create music. Music. The writing sings. It has a pleasant rhythm, a lilt, a harmony. I use short sentences. And I use sentences of medium length. And sometimes when I am certain the reader is rested, I will engage him with a sentence of considerable length, a sentence that burns with energy and builds with all the impetus of a crescendo, the roll of the drums, the crash of the cymbals—sounds that say listen to this, it is important.
So write with a combination of short, medium, and long sentences. Create a sound that pleases the reader’s ear. Don’t just write words. Write music.
As Provost says, you can create music with how you give out the sentence lengths. Dynamic! It has rhythm, cadence, and still follows the rules of the English language!
Hope that was helpful.