Ethernet comes up after every other suspend with Ubuntu 14.04

I've got this Optiplex 760 that I'm using with Ubuntu 14.04 and it's running really nicely. Ubuntu is a pretty slick setup and it works well and is very reliable.

But there's one thing that's driving me bats: My ethernet adapter disappears after every other suspend, forcing me to suspend again to bring it back.

I don't see that there's anything esoteric about my hardware, and it works fine once it comes back. Weird.

lspci
00:00.0 Host bridge: Intel Corporation 4 Series Chipset DRAM Controller (rev 03)
00:02.0 VGA compatible controller: Intel Corporation 4 Series Chipset Integrated Graphics Controller (rev 03)
00:02.1 Display controller: Intel Corporation 4 Series Chipset Integrated Graphics Controller (rev 03)
00:03.0 Communication controller: Intel Corporation 4 Series Chipset HECI Controller (rev 03)
00:03.2 IDE interface: Intel Corporation 4 Series Chipset PT IDER Controller (rev 03)
00:03.3 Serial controller: Intel Corporation 4 Series Chipset Serial KT Controller (rev 03)
00:19.0 Ethernet controller: Intel Corporation 82567LM-3 Gigabit Network Connection (rev 02)
00:1a.0 USB controller: Intel Corporation 82801JD/DO (ICH10 Family) USB UHCI Controller #4 (rev 02)
00:1a.1 USB controller: Intel Corporation 82801JD/DO (ICH10 Family) USB UHCI Controller #5 (rev 02)
00:1a.2 USB controller: Intel Corporation 82801JD/DO (ICH10 Family) USB UHCI Controller #6 (rev 02)
00:1a.7 USB controller: Intel Corporation 82801JD/DO (ICH10 Family) USB2 EHCI Controller #2 (rev 02)
00:1b.0 Audio device: Intel Corporation 82801JD/DO (ICH10 Family) HD Audio Controller (rev 02)
00:1c.0 PCI bridge: Intel Corporation 82801JD/DO (ICH10 Family) PCI Express Port 1 (rev 02)
00:1c.1 PCI bridge: Intel Corporation 82801JD/DO (ICH10 Family) PCI Express Port 2 (rev 02)
00:1d.0 USB controller: Intel Corporation 82801JD/DO (ICH10 Family) USB UHCI Controller #1 (rev 02)
00:1d.1 USB controller: Intel Corporation 82801JD/DO (ICH10 Family) USB UHCI Controller #2 (rev 02)
00:1d.2 USB controller: Intel Corporation 82801JD/DO (ICH10 Family) USB UHCI Controller #3 (rev 02)
00:1d.7 USB controller: Intel Corporation 82801JD/DO (ICH10 Family) USB2 EHCI Controller #1 (rev 02)
00:1e.0 PCI bridge: Intel Corporation 82801 PCI Bridge (rev a2)
00:1f.0 ISA bridge: Intel Corporation 82801JD (ICH10D) LPC Interface Controller (rev 02)
00:1f.2 SATA controller: Intel Corporation 82801JD/DO (ICH10 Family) SATA AHCI Controller (rev 02)
00:1f.3 SMBus: Intel Corporation 82801JD/DO (ICH10 Family) SMBus Controller (rev 02)

uname -r
3.19.0-47-generic