You say that ‘cycle’ seems too clinical, so that means that you'd like something slangish. At that, you don't give us enough information to devise something which would be consistent with the culture of your world.
However, I could walk through an example.
So, these people have cycles of sleep periods which differ enough from the cycles of daylight and darkness that they need their own metrics. Then begin by referring to them as just that — indeed, what we call them now: ‘sleep cycles’.
Of course, that's three syllables. In parlance, they prefer a short, brief single syllable that rolls out of the vocal apparatus. Some people began calling them ‘sleeps’, vis–à–vis
It's been three sleeps since then. Can't you let it go?
Indeed, some people already use it in that manner, more or less. However, it didn't really catch on in this fictional world: perhaps context wasn't enough to distinguish it from the homonym.
Then, someone noticed that another word for ‘sleep’ is ‘slumber’. Now, along it went, juggled about on the lingual pods of language; people couldn't exactly begin saying things like
Ten slums is more than enough time for you to reach a decision!
but they could, perhaps, say
Only nineteen bers ago, you told me that you couldn't stand the sight of me. Now, what — you want to follow me around like a anxious toddler?
Of course, the acceptance of that word depends on which vowel you use, but it seems as good a beginning as any to me.