If you really care about getting something done, then you're going to need to stay on it. The best way to do it without being annoying is:
1. To be extremely polite and articulate each time you follow-up as it will not only make people more receptive to in your inquiry, but will help them have a better understanding of what it is you actually need;
2. Try to secure a "when can I expect to hear from you?" ballpark figure from the parties involved; that way they can set the expectation for when you should hear back from them and you can politely hold them to it;
3. Give people a reasonable amount of leeway - if the person you're contacting is extremely busy, it might take them weeks to follow up on a simple email. I contacted a CEO of a company which represents a model customer for my own startup service in late March to get some feedback from him; I didn't actually end up speaking to him until yesterday. That's just how it goes with busy people some times - just stay organized, keep a list of all of the people you've contacted and when, and periodically remind them when enough time has passed.