Okay, thanks for the responses. I figured out what the deal was. For anybody who might have a similiar problem, this is what I did:
First, I went to Linksys' web site, and downloaded updated drivers for the LNE100TX 10/100 LAN Card.
Second, I had to change some settings:
Start>Settings>Control Panel>Network
Configuration>SELECTED Linksys LNE100TX Fast Ethernet Adapter(LNE100TX v4)
Properties>Advanced
APM Mode Wakeup:Disabled
Link Wakeup (Off->On):Disabled
Link Wakeup (On->Off):Disabled
Magic Packet Wakeup:Disabled
Media Type:AutoSense
Some of these were set to Enabled by default, I just switched them all to Disabled, and now my machine doesn't wake up on it's own.
One more thing of note: This NIC card came with a cable to connect to the motherboard to utilize the WOL (Wake-On-LAN) features. There IS a connector on my motherboard to support this, but I did not hook it up. Yet, it still utilizes WOL features. I thought that was kind of interesting/odd.
Thanks again for the input!