Disclaimer: I'm barely above layman when it comes to poetry, and not a native either.
So let's just say we're dealing here with poems that are not deliberately structureless, nor those that intentionally alter the expected structure.
Take any of Shakespearean plays and look at the lines. Count their syllables and where the stress of each word lies. While hardly universal across the board, the predominant structure is that of ten or so syllables, and stress falling on every second syllable (often reversed in the first syllable pair).
e.g.:
Now is the winter of our discontent
/ - - / - / - / - /
the famous line:
To be or not to be, that is the question
- / - / - / || / - - / -
has generally the same structure, but with a pause in the middle (indicated by the punctuation) and the second part altering the regular structure with an inversion in the first syllable-pair (it feels natural after a pause, just as it does in the opening syllable-pair), and what is called a feminine ending (an extra unstressed syllable).
Those changes are minor, and do not change the general rhythm of the reading, though. It's still five stressed syllables and a heartbeat-like rythm.
It's the iambic pentameter, and it's ubiquituous in S.'s plays. This is what sets the rhythm of reading - you automatically expect where to put the stress.
Do note, that for non-native English speakers the position of stress in a word is not always obvious. They might be eloquent erudites in writing but still carry old habits from their native language that makes them stress wrong parts of words (which is a part of what is called having a foreign accent).
So the lack of rhythm you may feel in reading English poetry might come from that, and there's little you can do about it bar studying phonetics of the language.
When you look at the sonnets (which often use the iambic pentameter as well), you'll see a higher-level structure in the way the lines "clump" - 4,4,4,2
and rhyme:
abab cdcd efef gg
So, once you see a poem with three four-line stanzas and one half as long, and somewhat longish lines, you can immediatelly expect it to rhyme like a sonnet, and have a rhythm of a line given by the iambic pentameter.
Other forms of poems will have different kinds of rhyming and meter, but you can generally expect it to be consistent across the poem. The first line will set the meter, and the first rhyme will set the rhyming scheme.
In other (usually old and middle English) poems you may find alliteration and caesuras that set the way you should read them: there's a pause in mid-line (caesura), and the key words on each side tend to start with the same letter, so you naturally put more stress on those. They tend not to rhyme.
e.g.:
Byrhtnoth spoke, lifted shield,
shook slender ash-spear, with words spoke,
angry and one-minded gave him answer
(from Battle of Maldon)
While the alliteration is mostly lost in translation, just insert a pause mid-line, and you have a nice, powerful rythm.