Check out Operation Smile. www.operationsmile.org
. They go all over the world, are non denominational, the cost is reasonable ($500) per mission, includes airfare and housing, and their time commitment is 10 - 14 days.
There are a lot of smaller trips run by universities and individual doctors. I would keep an eye on the news and get in touch with some of those people directly. Once you get on a trip, you get to know people, and its easier to get on subsequent trips.
I wouldn't worry about the religious part. You deal with and support other people's religious practices and needs wherever you work, whether you share their faith or not.