For "normal" file operations, taking an LVM snapshot of the mounted filesystem
and then making your backup from that should be sufficient. This should even
work for postgreSQL database files (though, it is not optimal). MySQL has a
history of being more flaky, but it might work there as well.
If you snapshot a mounted file system, the snapshot will be approximately
equivalent to the original file system, uncleanly unmounted at that exact
moment (think: power failure). It's possible to then take backups of an
active system with no downtime (although I/O load will certainly go up during
the backup). If you mount the snapshot as part of the backup procedure, a
journaled file system will want to replay the journal then. Otherwise, a
journal replay will be required at restore time. PostgreSQL (etc.) will also
end up doing a journal replay / recovery at restore time.