Sorry, most organizations do require payment (since they are normally not charging the people they are giving the service to and need to make money for other things).
I have gone abroad twice to volunteer/intern. The first time was study abroad with an internship in a health clinic/ "hospital." That was done through KEI and I did it over the summer. Since it is study abroad, I got scholarships and financial aid to help me. Speaking Spanish wasn't required, but I wouldn't do it if I didn't. Cost of 2 months (included 12 credits) was $6000, after airfare and vacations there $7800
Link:
http://keiabroad.org/
My program link:
http://keiabroad.org/programs/guayaquil/overview.php
Another time I went for just a week with an organization called Hearts in Motion. My school has a program set up with them that gives us one credit for doing it, so again we get financial aid and some of the doctors we went with give out scholarships. Cost of 10 days for program $900, after airfare $1700.
Link:
http://heartsinmotion.org/
I know it seems expensive, but for me they were totally worth it. Include those on scholarship essays later on and you will make back part of the money.