What's the best way to backup DB files via network on Linux and Solaris machines?
Well theoretically you can issue the command "alter tablespace begin/end backup". Before and after you execute rsync. You will also need a backup copy of the control file. This approach was used many years ago before RMAN was introduced. The restore really requires Oracle experience. This approach is for example described in this book: "Unix Backup and Recovery".

A really recommend you to get familiar with RMAN. Even if you do not use backup agent like TSM/Netbackup you can still create local disk backups using RMAN and then you can synchronize this storage using rsync.