As the page you linked implies, any point in time snapshot technique that includes the data files and the journal will suffice, LVM is just one option. EBS snapshots in Amazon EC2 will also work, as will similar snapshot solutions on SAN, NAS etc. You are not limited to LVM, but that is generally a solution people can implement themselves.
In terms of whether you can copy files to perform a backup, the answer is yes, but only if you stop all writes to the node you are backing up (thereby guaranteeing no changes to the files during the copy). You can do this in a couple of ways:
The most straight forward way is to just shut the node (this should be a secondary) down, copy the files, then start the node back up and let it catch up to the primary (check optime using
rs.status()). Rinse and repeat (if you wish) to cycle through all nodes in the set, though the nodes are all identical, so one copy should generally be enough.
The second way (mentioned by sysadmin1138) is to fsync (flush data to disk) and lock (prevent writes) the node but leave it running using the
fsyncLock command (again, this should be a secondary). Once you have completed the copy, you unlock the database using the
fsyncUnlock command. There are dangers inherent in this technique - for example (and particularly if you are using authentication), you should always lock and unlock on the same connection, otherwise you risk locking yourself out of the database and having to kill the process to recover.
As for other risks, it is common to use a hidden node for backups in each case, which prevents accidentally attempting reads from the node while it is behind, and/or while it is locked (depending on your methods).
Finally, there is one further (paid) option - MMS Backup. This service will essentially do all this for you (for a fee) and give you extras like point-in-time recovery and more - note: I work for MongoDB so I won't give you the hard sell here, but feel free to evaluate it yourself.