There is an interesting argument that none of the classical rhyme schemes is natural to English, and that instead alliterative verse is the most natural form. JRR Tolkien is well-known for his work on this theory (in addition to some other, more obscure works).
Alliterative verse is characterized by (1) the use of head-rhymes or alliterations and (2) meter based on accent, not on feet (accentual verse).
(1) Alliterations are based on sound, not spelling, and some sounds are considered to alliterate even if they are not identical. All vowels alliterate with each other. So center alliterates with sin, elf alliterates with antler, and victor can alliterate with fern.
(2) In meter, stressed syllables are counted, and unstressed syllables are not counted at all. So the following lines have 2 stresses per line, even though the first line has 4 syllables and the 2nd has 5.
Baa baa, black sheep
Have you any wool?
Much of the poetry of Anglo-saxon was alliterative verse, notably Beowulf. Its popularity as formal poetry waned after the influence of Norman French and classical forms. It is still apparent in modern forms like cowboy poetry and rap. Tolkien's translation of Sir Gawain and the Green Knight is a good example of alliteration in modern English. Here is an excerpt from Langland's Piers Plowman in modern translation:
Among them I found a fair field full of people
All manner of men, the poor and the rich
Working and wandering as the world requires.
The arguments in favor of its naturalness to English are
English does not have many inflectional endings, so end-rhymes are less natural than in languages that do use such endings.
English is a stress-timed language, not a syllable-timed language or a mora-timed language. Accentual verse matches the patterns of stress-timing more naturally.
There are some problems with the above analysis, to be sure: if alliterative verse is so natural, why isn't it more widely used? Is the stress/syllable/mora-timing distinction even real? However, I thought that this question does merit a discussion of this lesser-known but important verse form.
Natural verse is very vexing
To define: Dante’s Divine Comedy
In terza rima’s rhythmic mode
Apprehends Italian’s essence perfectly;
The dactylic hexameter of Homer’s distichs
Gave us Greek and Roman rhymes,
The prosodic feet which our fathers professed
As the Classical model for modern poetry;
But the uncouth consonant clusters
Of Anglo-Saxon speech are served
Better by a blunter form;
Alliteration loves our letters' patterns!