Due to certain browsers handling HTML5 differently, we also need MP4 versions of the movies in addition to OGG. So, I used HandBrake (specifically HandBrakeCLI) to batch convert all of the MOV files to MP4. Two commands this time - one to convert and one to rename:

find media -name *.mov | xargs -I file HandBrakeCLI -e x264 -q 20.0 --input file --output file.mp4

find media -name '*.mov.mp4' | xargs rename 's/\.mov.mp4$/\.mp4/'

Be sure to specify "-e x264" when using HandBrakeCLI or else it'll use ffmpeg.