NOTE: I have since gotten an up-to-date ffmpeg (not the outdated version of avconv in the Ubuntu repositories), and now shlck's
-map_metadata -1 works perfectly. If you can, use that solution instead (less typing). I'm leaving this here for anyone using an outdated version.
The easiest way to do this is to set
-map_metadata to use one of the input streams, rather than using global metadata. 99% of the time this should work. NOTE: I'm using avconv, because that's in the Ubuntu 12.04 repositories; this will probably be drop-in compatible with ffmpeg, since their syntax always is in my experience.
avconv -i input.mp4 -map 0 -map_metadata 0:s:0 -c copy output.mp4
This will take the metadata from the first data stream (normally the video stream) and use that to replace the global metadata of the container file. This works because most of the time, the data streams have no meaningful metadata written to them; however, sometimes they do, and you want to completely get rid of that metadata. Unfortunately, the only way I can think of to do this used a pipe and two avconv processes.
avconv -i input.mp4 -f wav - | avconv -i - -i input.mp4 -map 1 -map_metadata 0 -c copy output.mp4
This takes advantage of the fact that WAV files can't contain metadata (since the format was created before metadata tags existed).
Both of these methods blanked all metadata on a file I just tested them on - all that
exiftool reported on was the codec information, and avprobe reported no metadata to me. Using a pipe for this is pretty ugly, and the first method will work in 99% of cases, so that should be preferred.