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).
What you're describing is standard procedure for any email marketing (outbound) tool. Someone already mentioned MailChimp but so many others. You'd set up rules based on when and who to send reminders after X number of days if some pre-defined action didn't happen (open email, register for something, etc.). These tools also track bounces, open rate, unsubscribes, etc.
You should be diligent in only sending additional emails to a sub-set of the initial list. It'd look pretty bad to send a reminder email to sign up to someone who already signed up.