There are many of these services around. I haven't used any myself, but I did once investigate them for a client. Among those that I shortlisted were Twilio SMS, Neximo, and Clockwork SMS.
Like Mailchimp, many of these services offer a limited free version, and a more useful paid-for version. Be aware also that some of these services are country-specific (at least, that was the case when I last investigated them).