The Day of the Dog
This category has a calendar, and it is the thing to understand before visiting.
The traditional visit is made on an inu no hi, a Day of the Dog, in the fifth month of pregnancy. The twelve-day zodiac cycle means one comes around roughly every twelve days. Dogs are associated with easy births, which is the whole logic. On that day the expectant mother receives the hara-obi, a maternity sash, blessed at the shrine.
If you visit Suitengu in Tokyo on a Day of the Dog you will find it packed, the number of people admitted to the hall capped, and accompanying family potentially turned away. If you visit on any other day you will find it quiet. Both are worth doing; they are simply different experiences, and the shrine publishes its calendar so you can choose deliberately rather than by accident.
The gesture people get wrong
Suitengu's kodakara-inu is a statue of a mother dog with a puppy, ringed by the twelve zodiac animals. Visitors stroke the dog constantly. That is not the practice.
The shrine's own instruction is to stroke the zodiac animal of your own birth year, holding your wish in mind. Not the dog. It is a small thing, and it is the single clearest example we have found of a documented practice being displaced by an obvious-looking guess.
A note on visiting
This is a category where foreign visitors sometimes feel they are intruding, and the honest answer is that the shrines do not treat it that way — Suitengu's grounds are open to anyone and the statues are public. But it is worth registering that people around you are there about something consequential and often anxious, and behaving accordingly.
Formal prayers at Suitengu cannot be booked in advance. Same-day reception only, and once the cut-off passes they will not take you.
