Using GNU sed (on Linux or Cygwin):
# Removing BOM from all text files in current directory:
sed -i '1 s/^\xef\xbb\xbf//' *.txt
On FreeBSD:
sed -i .bak '1 s/^\xef\xbb\xbf//' *.txt
Advantage of using GNU or FreeBSD sed: the -i parameter means “in place”, and will update files without the need for redirections or weird tricks.
On Mac:
This awk solution in another answer works, but the sed command above does not work. At least on Mac (Sierra) sed documentation does not mention supporting hexadecimal escaping ala \xef.
A similar trick can be achieved with any program by piping to the sponge tool from moreutils:
awk '…' INFILE | sponge INFILE