Old habits are hard to break, so it may take awhile for your children to understand that you are serious and are no longer being controlled by their emotional outbursts. Keep your cool and continue about your day, not letting them see the frustration you feel. And always, always, go through with the outcome that you have described to them. Be consistent.
Most importantly, when the make the right decision, be sure to give them a big hug and let them know how proud of them you are.
See also:
Creating a Discipline Process at Home
When is a Tantrum Just a Tantrum? Or is it Time to get Help?
