Welcome to English Forums, Steve.
I don't think
Dear Sirs is old-fashioned; it is a set phrase and still heavily in use. On the other hand '
to whom it may concern' is not really appropriate for business letters-- it is a formula used in testimonials mostly, and is rather old fashioned.
Still, I see the awkwardness of using the male gender in this day and age. If it is a very formal business letter, I would stick with
Dear Sirs, but otherwise something like
Dear ABC Production Department might be the better part of valour; personally, that is the way I greet in most e-mail business letters.