The observed phonology of Klingon
We will begin by noting that The Klingon Dictionary lists 21 consonants and five vowels. The consonants are b ch D gh H j l m n ng p q Q r S t tlh v w y ', and the vowels are a e I o u. Without further embellishment to that classification, here's what we observe about the phonological structure of Klingon syllables.
With the single exception of the endearment suffix -oy, every prefix, suffix, and single-syllable word in The Klingon Dictionary begins with a single consonant followed by a single vowel. The syllable may end there (as with a handful of words and every pronominal prefix), or it may conclude with either a single consonant or one of the three consonant clusters rgh, w', or y'. w never follows o or u. The individual syllables making up more complicated words all follow this same pattern.
Denoting consonants with C and vowels with V, here's what we find as valid syllables: * oy (only as the Type 1 noun suffix) * CV * CVC (except for C_ow and C_uw) * CVrgh * CVw' (except for C_ow' and C_uw') * CVy' That's it.
The only two known exceptions to this in canonical sources are the names qIrq and pIqarD. Both are tlhIngan Hol representations of non-Klingon names.