First you upload a copy of the site at the other host and make sure everything is working properly. Hopefully media temple has the use of mirrors so you can view your site before you change the dns settings of the url so you can check that everything is working. Once you are sure everything is fine THEN change the dns settings. So long as you wait before removing the files on dh for 24-48hours there should be little, if any, noticeable downtime while the new dns settings are propagating.
I hope you got paid first btw.