English syllabic poetry typically uses a straight count: you count the syllables. This is not because this is the most natural way to do syllables, but because number of syllables is, along with stress, about the only thing about syllables that's reasonably stable in English. Even then there are plenty of puzzles, like the 'ire' family of syllables: briar, buyer, choir, dire, fire, gyre, hire, higher, ire, lyre, liar, mire, pyre, sire, tire, wire, and so forth. In some dialects there are differences among some of these, but in some dialects (my own, for instance) these are all exact rhymes, and they are all capable of functioning as one or two syllables. The reason is that with a strongly accented vowel sound followed by a weakly accented vowel sound, with no consonant or semi-consonant to distinguish, the strong vowel can dominate the weak vowel to varying degrees, ranging from completely dominance (so that you only hear the strong vowel) to carefully enunciated nondominance. And the weakly accented vowel sound remains pronunciable because of the 'r' at the end, which, so to speak, gives it something to hold onto. This allowance for fuzzy syllable distinctions is only one of many ways in which English is sloppy with its syllables. Syllables can vary in length from dialect to dialect; there is no standard set of syllables, since English borrows syllables from all over; and so forth.
Some languages, however, are much more careful with their syllables. In such languages straight-count syllabic poetry is usually not very interesting, and is not all that distinguishable from prose; it would be like writing poetry in English in which every line has the same number of words:
The rabbit sat
on the carpet
with the dog
and the parakeet
while Jane called
John at home.
But with a relatively orderly syllabic language, there's more to work with, and the most important of these is the mora. A mora is a unit of sound indicating the extension of the syllable or, as it is usually called, the syllable weight; the word literally means 'delay'. To determine the moras you have to analyze the syllable into its parts. Every syllable has a nucleus, which is the sound (usually a vowel, although there are consonants in many languages, like m, n, and r, that can do so as well) that carries the syllable; many syllables also have a coda, which is the final consonant sound and an onset, which is an initial consonantal sound. Wikipedia actually has a nice article on the subject, and gives a typical set of rules for determining moras:
1. A syllable onset (the first consonant or consonants of the syllable) does not represent any mora.
2. The syllable nucleus represents one mora in the case of a short vowel, and two moras in the case of a long vowel or diphthong. Consonants serving as syllable nuclei also represent one mora if short and two if long. (Slovak is an example of a language that has both long and short consonantal nuclei.)
3. In some languages (for example, Japanese), the coda represents one mora, and in others (for example, Irish) it does not. In English, the codas of stressed syllables represent a mora (thus, the word cat is bimoraic), but for unstressed syllables it is not clear whether the codas do (the second syllable of the word rabbit might be monomoraic).
4. In some languages, a syllable with a long vowel or diphthong in the nucleus and one or more consonants in the coda is said to be trimoraic (see pluti).
In haiku what you are counting are not the syllables but the moras of the syllables: five moras, seven moras, and five moras. From what I understand, in Japanese light syllables (monomoraic) are fairly common, so there will often be a one-to-one correspondence. But there often will not, and Wikipedia notes that Tokyo, Osaka, and Nagasaki all have a different number of syllables but the same number of moras.
I imagine you could write genuine haiku in Cherokee and get a nice result; English not so much. For that matter, since moraics underlie length of vowels, you might be able to do so in Latin and Greek, although given the way the two form words I imagine that it would take a certain amount of epigrammatic genius. Latin and Greek metric poetry is a distant cousin of haiku -- and, contrary to common belief, we don't typically write genuine metric poetry in English, either, since one of the ways English is not careful about syllables is in the length of vowels. Instead we write rhythmic stress poetry whose patterns of stress are somewhat analogous to metric patterns.
But all this, of course, still leaves the question: despite the obstacles, can one write poetry in English that consists of three lines of five moras, seven moras, and five moras, that makes sense, and that sounds interesting? Is this a poem form that really requires a different sort of language from English? Or can it be done? One can write terza rima in both Italian and English, even though Italian is massively better suited for it due to its greater regularity of word-endings; can one write haiku in English as well, despite the extraordinary chaos and disorder of the English language? I have no clue.