Networking with whatever non-profit you are interested in, they would know others in their field looking for people.
http://techsoup.org in the forums there’s a career opportunities section. Pay is no where near the rate of private sector, but there is a better chance for longevity and doing something you like.
In addition to these sites, start volunteering for local/nearby charities. You'll find that it's a good way to build up your network (why do you think so many execs are on nonprofit boards?) and your credibility within the nonprofit space.
Why not use the usual job searching sites with the keyword of "non-profit" - most job descriptions include at least a blurb of the company and non-profits will likely mention their status.
....yes this is a joke