There's no need to be embarrassed. There's no obligation to say anything and you don't have to use "Say cheese". That is what people say to make the people in the photograph smile - it's difficult not to put your mouth in a smiley shape when you say the word "cheese" (or any other word with "ee" in it).
There's nothing wrong with "One, two, three" if your subjects knew that on/after "three" they had to smile or do something else all at the same time.
And as you can see from
this piece, actually asking your subjects to say "one, two, three" gets the same result as "cheese" because it has "ee" in it.