If I wanted to suggest a game idea to a company, how should I do that?

I tried once, they sent me a letter saying, basically, "It takes a lot of people to make a game and your suggestion is appreciated, but we really don't need your help". Try writing to them via snail mail, they're mailing addresses are usually on their web sites or on the boxes of video games.

Most companies do their best to even avoid reading such suggestions because if they have anything vaguely similar in the works then they get accused of plagarism.