An opt-in mailing list system such as GNU mailman would reach the people who want to be reached and give them plenty of options (e.g. individual mails versus digest form, the ability to turn it off temporarily for vacation, etc.).

I would strongly recommend you not automatically E-mail your users on a weekly frequency! If you must do this, make it infrequent (monthly, no sooner) and make sure you include genuinely useful information. A good example is LinkedIn, which includes information that I can definitely use (new developments among my directly connected peers), and they only send it if I haven't visited the site in about a month.