Among the important issues are getting experience in a related, or "parallel" line of work, such as programming or art; noting that there are few available design positions available, and these are typically filled from within the company; and remembering professionalism, courtesy, responsibility, confidence and diplomacy as important assets for any employee.
This blogger once worked for Atari as an artist first, then moved into game design to work on a title that ended up being scrapped in its eleventh hour, right before the company laid everyone off (funny story, remind me to tell it to you all sometime). When we were in the position to hire, another important factor was "Does this person have a well-rounded background?" Believe it or don't, it's important to have more to talk about than just games. Inspiration can come from any angle, and a wealth of general and specific knowledge in many different fields can be just as important as the ability to talk about what makes games fun.
Read through the whole piece on the CO site; it's a good read, and may help in crystallizing the dreams of someone reading this site.