The headers/cables of that Tsunami Drea case pertain to
AC'97 Audio connectors; the motherboard pinout you posted pertains to
HD Audio pins. HD Audio (released in 2004)
is NOT backward-compatible with AC'97 (an old standard), despite both having the same 10-1 physical pinout -- but they are electrically different.
Kindly check your motherboard manual
if AC'97
is supported. If it is, you should be able to select AC'97 in BIOS or Audio Software (usually the default is HD Audio).
On another note, does you case have the HD Audio headers aside from the AC'97 (usually on older cases, the AC'97 cables are piggy-backed to the HD Audio)?