I was uncomfortable with "networking" for many years, until a friend turned it on its head for me by his own example. This is the most connected guy I know: no matter what or who you need, he can get you there, or close to it.
So how does he do it? Simple: he is always on the lookout for ways to help other people. He pays no attention, in the first place, to what they could do for him -- that side of the equation takes care of itself. The other secret to his success is that he lets people help him; when he needs a hand, he'll ask for it. Letting someone help you out creates not just an obligation to reciprocate, but a particular kind of bond predicated on the way we all feel good about having been able to help.
To accept help is, in and of itself, to give a gift. In the end, the "network" that grows out of this "method" is robust, productive and not in the least bit sleazy or manipulative.It made me feel a lot better about the whole thing, in the sense that I thought "yeah, I can do that."