Non-Religious International Activism/Charity/Political Organizations?

There are plenty of charities which have or stem from a religious motivation, but whose activities don't involve promoting religion or practising indoctrination. Oxfam would be an example, as would Concern or the Samaritans. (Some people would put Amnesty International in this category also, on the basis that its founder was motivated by religious belief.)

So you need to ask yourself are you more concerned about what the charity does, or about why they do it?

As the Amnesty example shows, if we are looking at motivation there isn't really a clear dividing line between religiously-motivated charities and those with a purely secular motivation. The motivation of those involved in a charitable organisation can change over time and furthermore the "official" stance of the organisation is not necessarily the same as what motivates all of those involved. I know people working for Catholic development agencies, for example, who have no religious belief and are generally hostile to the church, and they don't feel that what they do in those agencies is at all inconsistent with their beliefs and values, even though they are explicitly Christian agencies. Conversely there could well be people in secular agencies whose motivation for involving themselves is a religious one.

My inclination is to look at what people do, not at why they do it. But your mileage may vary.

If you're looking for agencies with (so far as possible) a completely secular background and ethos, in addition to UNICEF and Amnesty, you could consider Médecins Sans Frontières and the Irish Red Cross . There's also Save the Children and, of course, GOAL.