I would like to dispatch certain emails to users that sign up for my email communications at a fixed timeframe. For example, send email A after 7 days of signing up, email B after 21 days, etc. (assuming they don't unsubscribe). Are there email marketing services that can accommodate this?
MailChimp can do this. You can sort of see it in the first screenshot image there.
Additionally, if your subscription form has any date-based custom fields, you can use those as the basis for automated scheduled messages, like a "happy birthday" note(assuming you collect birthdays, obviously).