Is that possible to backup MongoDB sharded cluster without LVM or mongodump? There is a "LVM-solution" in MongoDB documentation: http://docs.mongodb.org/manual/tutorial/backup-sharded-cluster-with-filesystem-snapshots/

Can I do the same thing, but without using LVM snapshots, simply copy the files of my secondaries, and include a "mongodump" of my config data?
The LVM method works they way it does because snapshots are really fast, and the outage is really small and easily within the replication-log window. Using file-copy methods extends how long the replica-member is not replicating, so how long you can take to back up is constrained by the maximum replication-log allowed on the master node.

Secondly, snapshotting takes a point-in-time backup of all files at the same time which improves the consistency of the backup. For a file-copy style backup the files will be read sequentially, so the use of db.fsyncLock() is massively important; if you don't, the chance you're backing up inconsistent files is really high. It's the use of db.fsyncLock() that starts causing the replica to fall behind and the replica-log to increase on master.