Intuitive understanding of frequency
It's about units and dimensions. And I'm probably not the guy to cast an answer on that but I would like to state it as follows. In short; length, time, mass etc. are dimensions, whereas meter, seconds, kg are units of them respectively in SI. (for other dimensions and units, a physics book might help.)

Now, if a wheel rotates 7 times in 2 seconds; it's rotation frequency $f$ is found as:
$$ f = \frac{7 ~\text{ cycles}}{2 ~\text{ seconds}} = 3.5 ~\text{cycles-per-second (cps)}.$$ 

As can be seen, frequency is given by a (cycle) number divided by time. The cycle number has no physical units or dimensions; it's just a repetition count. Whereas the duration time has a unit of seconds (in SI).

Thus the dimesion of frequency is 1/time, whose SI unit is $1/s = s^{-1}$. This unit is also, and more frequently, denoted as Hz where it (implicitly) stands for (cycles) per second indeed. So, $3.5$ Hz stands for $3.5$ cycles per second.

Now, similarly the period is defined as the duration time per one cycle. Again note that since cycle is a number and does not posses a unit, then the dimension of the period is time and its unit is seconds in SI. But you can always denote it also as seconds per cycle, rarely practiced though. 

Finally, using both notations you have 
$$ T = \frac{1}{f} $$ 

So if frequency $f$ is defined with a unit of Hz ($s^{-1}$), then the period will have a unit of seconds (s) in SI. Otherwise if you define frequency as cycles per second then the period unit will be seconds per cycle.

So if you define the frequency as $3.5$ cycles-per-second , then the period associated is:
$$T = \frac{1}{f} = \frac{1}{ 3.5 \frac{\text{cyc}}{s} } = \frac{2}{7} \frac{\text{s}}{\text{cyc}}$$ 

Otherwise, in accepted norm, as was you would define freqquency as $3.5$ Hz. its period in SI will be:
$$ \boxed{ T = \frac{1}{f} = \frac{1}{ 3.5 s^{-1} } = \frac{2}{7} ~\text{ s} } $$