I got the idea from here. He uses a Python script and the text interface is not free. I thought someone must have already thought of this for R. There is an easy solution. You can send an email fairly easily in R. You can also send text messages as emails. This page gives a few examples of how to send an email in R. Sending a text via email will depend on what carrier you have and I am sure standard text message fees apply.You can only send 20 emails a day with this package, but that probably won't be a big deal.
It is basically a one-liner. Here is the code I used:
subject="Notification from R",message="MCMC Finished", password="rmail")