If they say not to call, don't call. I would not be happy AT ALL if everyone who applied started calling me about a job opening. That would mean more than 100 people bugging me when only 6 or 8 of them will even get phone screens by HR. Multiply that number by all the positions that HR is recruiting for and, more likely than not, you'll just piss somebody off.
That said, if you know anyone who works at the company where you're applying or you have some sort of in, you should definitely use those leads. While I don't want random people calling me, I appreciate referrals by people I trust. Use LinkedIn to your advantage.
Unfortunately, Midgi's right to a certain extent. I hired for two marketing coordinator positions simultaneously last fall and we advertised them externally. However, I never even interviewed any external candidates. I had an overwhelming response internally from people who didn't even work in the marketing department. They had mostly taken customer service jobs "to get by" during the economic downturn so they had the trifecta of experience--marketing, our company, and the industry. Win-win-win.