I always feel sorry for you aspiring Game Designers who want to break into the business. Why? Because you are going to have a hard time - a VERY hard time.
That's not to say that things are impossible - they certainly are not. With hard work and dedication, anyone can become a game dev no matter what discipline you come from. But the fact remains that companies are very suspicious about junior Game Designers as your kind holds a lot of responsibility - and power - within a game team.
1) Acquire as many hard skills as possible!
You don't need to become an expert on code or 3D, but by knowing at least the foundations of code/computer science and things like art production or level design, you are on a good way of becomming a Jack-of-all trades - which is a very good quality to have as a game designer, especially if you apply for smaller studios where multi-talented people are favored over experts within just a single field.
2) Do as many small projects as you can, and avoid larger projects. Anything larger than 10 weeks full-time should be scrapped. When presenting these projects on a portfolio, employers are interested mostly in how your thinking was, what problems you faced and how you dealt with them. Showing off mechanics or graphics is unimportant. Employers are very cautious about hiring junior game designers unless they have solid evidence that they can work through all the problems that comes up for a designer during a project.
3) Don't be afraid to apply for and take a job as QA (quality assurance) tester. For many designers, this is a perfect way of getting a foot into the industry and get a foundation that you can work from and improve. QA testers are the last line of defense (compared to Tech Artists who are the first line of defense) in a game production, as they have the responsibility of making sure that no bugs or gameplay defects are shipped out to the customers. As a QA you will get insights into code as well as art and level design which in turn will aid you in becomming a better designer.
4) The second most popular way of becomming a professional Game Designer is to go the indie route and start up your own company. In this case, it is even more important that you are a Jack-of-all trades. Find people who are just as multi-facetted as you and start producing. Examples: don't join forces with people who are too caved-in on just one discipline - like a concept artist who has no experience with game engines, or a programmer who comes from a background as a network engineer.
5) Never be afraid of tossing away an idea or a prototype if the gameplay is NO FUN. I've seen so many smaller studios push for an idea that simply ain't fun or competitive on the market, because of fear of failure - or they may think that they cannot afford cancelling a production. FUN trumps everything else in a game.