My opinion (and I think there has to be a large element of that) is that the presence, and ultimately identification of the orbital period, of exomoons is going to come from very precise transit photometry.
If a sufficiently large moon orbits a planet, then this will leave its signature in the transit light curve. A "Fourier-type analysis" of the light curve might then reveal a periodic nature to the light curve shape during transit that could be attributed to the orbit of the moon. The requirements here would be a sufficiently large moon that its relative position alters the transit shape. The exoplanet also needs to be in a short period orbit so that you can observe lots of transits.
A related (and better) technique, that in practice would be used simultaneously, would be to look for the signature of a moon using changes in the duration and timing of transits that recur in a periodic way. The exoplanet-exomoon barycentre is what follows a Keplerian orbit; but the area-weighted "centre of opacity" of the system will, in general, not coincide with this barycentre because mass is proportional to $r^3$, while obscuring area is proportional to $r^2$. This will lead to a wobbling of both the time of transit
and the transit duration that may again yield a periodic signal which could be identified with the exomoon period. These methods are discussed extensively by Kipping 2009, who point out that the transit timing and transit duration signatures have different dependencies on the exomoon mass and separation from the exoplanet and thus a careful measurement of both could yield the exomoon mass. i.e. Each method in its own has a degeneracy which means the exomoon mass/period could not be determined, but measured together, the degeneracy is broken.
There is a trade-off here. A wide exoplanet-exomoon separation will give bigger signatures, but of course by Kepler's third law, the orbital period will be longer and so you would likely need a longer dataset of transits to identify any periodicity and pin down the amplitude of these signatures.
At this stage, I don't think there is any likelihood of hearing about an exomoon period/mass in the next few years. Exomoon detection by these methods is though a possibility and may already have occurred (see Teachey & Kipping 2018) . Perhaps the PLATO mission, which will produce better light curves than TESS or Kepler, and have long datasets will stand a better chance (and indeed exomoon detection is one of the mission goals - see Rauer et al. 2014).