Portfolios are a great tool if, and only if, they can grab the person and make them say WOW. Otherwise you may be better off not using one.
It sounds to me as if the work you do is more translating other's ideas into reality rather than doing the design yourself. So possibly you got judged on the skill of the art director and not your own skill of moving from design to working model. It is possible the people who were not happy with your portfolio were looking more for a design person.
Recognize that not even the best portfolio is going to impress everyone. And you are going to be in competition with others who may have a portfolio that is more in the style the person is looking for. Anything creative has styles that work for some people but not others.
You answered part of your own question when you said they seemed to want some actual site they could go to to play around in. While you clearly can't offer up your employers site for that, what you can do is set up a personal site just for portfolio purposes. See if you can get a designer to help you out with a design (someone just out of school (or a senior) who could use some additional things for his portfolio would be perfect) and then show step by step how you translate that design to a working model. Then you have a working model to go through with them and a process available to show what you did to get there.
Once you have created your sample site, then get some trusted friends to look at it as if they were a hiring manager. It's best if you know some hiring managers (maybe at an earlier job) who can critique for you. Get some of the art directors you know to critique for you. If you do all this now while you are not looking for a job, you will feel less pressured and can take the time to get it right. Then as your skills change and increase, update your sample site to reflect those skills.