I'll give you an example of a good relationship that you may have already read on a previous post of mine: A foundation with whom we have a good relationship was meeting in December for their final review of 2004. It turned out they had some "spare change" in excess of $1 million they hadn't granted last year. They called our VP of Advancement and asked for a proposal. While they didn't grant us $1 million, they did grant us $100,000 and agreed to review the $1 million proposal again at their next meeting. They didn't call the Museum of Fine Arts or a local hospital or the United Way or the Make A Wish Foundation. They called us. That's because we have a good relationship with them.
I've been at my current institution for over two years now. In that time I have seen numerous grants approved for between $500,000 and $3 million. Do you honestly believe if you send a cold-call, shotgun letter that you're going to get such grants?
And by "good relationship" I mean our Board members and a Foundation's trustees live in the same neighborhood, their kids go to school together, their families exchange Christmas cards. I get jokes and family pictures forwarded to me by Foundation officers. Our alumni graduated with their Trustees. It's a normal business relationship; if you are acquainted or friendly with someone, it makes business easier to conduct.
Nobody can tell you how to build a relationship. But you have to want to do it. It sounds to me like you're not even serious. If you are, try picking up the phone and calling. Introduce yourself. Tell them you're interested in learning more about the foundation. Don't even MENTION your organization (unless you're asked, of course). If you meet someone at a social event who you learn is affiliated with a foundation, give them your card. Learn about their interests. Just be friendly.
Or sit at your desk and post messages on a message board. Whatever floats your boat.