Power management has come a long way from where it was a few years ago. In my experience, cheap chipsets often do a bad job on implementation. Companies like VIA and SIS are cheap, but you get what you pay for. I've had a lot of success with Intel chipsets on both Intel and Asus motherboards. The one and only VIA (its an Asus motherboard) chipset I ever had works poorly.
Hmm, that is very strange that it would work after shutdown but not after hibernate. From a hardware standpoint, those two states are identical. The only difference is that a hibernate file is stored on your harddrive that windows uses to restore the previous memory contents. But the hardware knows nothing about that.
The first thing you should do is make sure you have the latest BIOS for your motherboard. From your post, it sounds liek your network device is integrated on the motherboard. It is BIOS responsibility to properly save and restore device settings on power state transitions. Once that is done, go through your BIOS setup menu and look for setting to enable WoL. For Shutdown and hibernate modes, BIOS is in (almost) complete control.
For standby (and to a lesser degree hibernate), your best bet is to make sure you have the latest drivers for your network device, then make sure it is set up from within windows. From device manager, open the properties page for your network device. There shoud be a "Power Management" tab with a checkbox that says "Allow this device to bring the computer out of standby." Realize that this setting will ONLY affect "standby" and has nothing to do with the "hibernate" or "shutdown" states.
I hope that helps. I work for a large computer hardware company and have a good bit of experience with this stuff in general, but not with your specific hardware. Let me know if anything is unclear or you have any other questions.
-Lou