How to figure out the rhythm of new materials (poems, rhymes, etc)?
You do not have to and cannot figure out the rhyme of the majority of modern Chinese poetry.
Because the rhyme is not necessary for it.

新诗 can be classified into the following styles according to the Wikipedia.


  3.1 分行诗 (lined format)
      3.1.1 自由诗 (free verse) the most popular style
      3.1.2 格律诗 (verse) out-of-date
  3.2 分段诗 (散文诗 prose poetry or prose verse) out-of-date
  3.3 图象诗 (calligram) out-of-date


The Wikipedia also mentions that 自由诗的特点是没有固定的结构、节奏，也不一定要押韵。
Free verse is characterized by the absence of a fixed structure and rhythm, and does not have to rhyme.

So, don't bother yourself about it.