Research Experience for Undergraduates: Summer Programs (that accept non-American applicants)

Here are some summer research opportunities in the US that fund non-US citizens. (I realize that you were asking for programs outside the US, but you framed your question in terms of being disappointed about having to fund it yourself so I thought this list might be helpful)

(I'm new, so the system won't let me post more than one hyperlink)

Research in Industrial Projects for Students (RIPS) 2011 at UCLA (Also has a program in HongKong)

http://math.williams.edu/small/application-information/ (their website says they have funded non-US students in the past)

NIMBios 2011 Summer Program National Institute for Mathematical and Biological Synthesis (NIMBioS) at the University of Tennessee, Knoxville

DIMACS REU DIMACS/DIMATIA REU CCICADA REU Rutgers Math Department REU (foreign students enrolled at a U.S. university are eligible under the CCICADA program)

Mathematical and Theoretical Biology Institute MTBI Summer Research Program

As an undergraduate in the UK, I did a summer research project in analytic number theory that was funded by the EPSRC Vacation Bursaries Scheme. It involved working under the supervision of an advisor on an individual project for approximately 8 weeks, and was a really good opportunity to gain research experience. I'm not sure what the constraints for eligibility are regarding nationality/citizenship, but the website I've linked to provides information about participating universities, who might have further information.

There is a similar scheme specifically for mathematics funded by the LMS