Don't go straight to du /. Use df to find the partition that's hurting you, and then try du commands.
One I like to try is
# U.S.
du -h <dir> | grep '[0-9\.]\+G'
# Others
du -h <dir> | grep '[0-9\,]\+G'
because it prints sizes in "human readable form". Unless you've got really small partitions, grepping for directories in the gigabytes is a pretty good filter for what you want. This will take you some time, but unless you have quotas set up, I think that's just the way it's going to be.
As @jchavannes points out in the comments, the expression can get more precise if you're finding too many false positives. I incorporated the suggestion, which does make it better, but there are still false positives, so there are just tradeoffs (simpler expr, worse results; more complex and longer expr, better results). If you have too many little directories showing up in your output, adjust your regex accordingly. For example,
grep '^\s*[0-9\.]\+G'
is even more accurate (no < 1GB directories will be listed).
If you do have quotas, you can use
quota -v
to find users that are hogging the disk.
