perl -i.bak -pne "s/^whatever it is your line to delete$//" files
Will create files without your line with extension .bak. Try it on one file and if you are confident enough to overwrite all your files you can do it by:
perl -i -pne "s/^whatever it is your line to delete$//" files
This will overwrite the original file. Other similar options are:
perl -i -ne "print if !/^whatever it is your line to delete$/" files
perl -i -pne "next if /^whatever it is your line to delete$/" files
Make sure you define the line from '^' (begining of line) to '$' (end of line) otherwise you might delete similar lines that have a partial match.