What method is the most script-friendly for moving files between servers?
The only methods I know of are SSH and rsync, but I don't want to jump into them if there is something else that is a standard.
EDIT:
The servers in question run on Red Hat Enterprise and Centos5
scp, sftp, rsync, are probably as close to standard as you get if you don't want to mount the remote filesystem.
If you are the same network, it would be somewhat common to also do an nfs mount of the remote system. Though if you are only infrequently moving files, this probably won't make much sense.
There are as many ways to do this as stars in the sky. Well maybe not that many but a lot.
If I had to pick one I'd probably pick scp with an ssh-keypair. I'd be able to give a more specific recommendation if I knew more about your use case.
I use Perl scripts based on the Net::SFTP::Foreign module. I find that this combination is easier to code for error detection, email reporting, file permission control, and fine grain attribute settings. The list of features this approach provides is extensive and can be viewed here:
http://search.cpan.org/~salva/Net-SFTP-Foreign-1.69/lib/Net/SFTP/Foreign.pm