I liked Martin Beckett's solution but found that file names with spaces could trip it up (like who uses spaces in file names, pfft :D). Also I wanted to review what was matched so I move the matched files to a local folder instead of just deleting them with the 'rm' command:
# Make a folder in the current directory to put the matched files
$ mkdir -p './matched-files'
# Create a script to move files that match the grep
# NOTE: Remove "-name '*.txt'" to allow all file extensions to be searched.
# NOTE: Edit the grep argument 'something' to what you want to search for.
$ find . -name '*.txt' -print0 | xargs -0 grep -al 'something' | awk -F '\n' '{ print "mv \""$0"\" ./matched-files" }' > doit.sh
Or because its possible (in Linux, idk about other OS's) to have newlines in a file name you can use this longer, untested if works better (who puts newlines in filenames? pfft :D), version:
$ find . -name '*.txt' -print0 | xargs -0 grep -alZ 'something' | awk -F '\0' '{ for (x=1; x<NF; x++) print "mv \""$x"\" ./matched-files" }' > doit.sh
# Evaluate the file following the 'source' command as a list of commands executed in the current context:
$ source doit.sh
NOTE: I had issues where grep could not match inside files that had utf-16 encoding.
See here for a workaround. In case that website disappears what you do is use grep's -a flag which makes grep treat files as text and use a regex pattern that matches any first-byte in each extended character. For example to match Entité do this:
grep -a 'Entit.e'
and if that doesn't work then try this:
grep -a 'E.n.t.i.t.e'