I found the solution, so I'm posting it here for anyone else that runs into this problem. This solution is for for those using a Dell 2007fpb monitor, and the Retropie install from ArcadeModUp.
#1 Don't bother trying to use an HDMI -> DVI adapter, some monitors will immediately enter Power Save mode no matter what you do. Use an HDMI -> VGA adapter instead and save yourself the trouble.
#2 You need to set the correct resolution on boot.
Connect a USB keyboard to the Raspberry Pi and press F4 to enter the linux shell. Type:
cd /boot
sudo nano config.txt
Find & change these 2 lines to:
hdmi_group=2
hdmi_mode=51
Hit Ctrl-X, and Y to save the file.
#3 The Dell 2007fpb monitor will ONLY work with 1600x1200 resolution. Since there is no reason to support any other resolution just edit your global retroarch.cfg file and set it there for everything.
Here's how:
Connect a USB keyboard to the Raspberry Pi and press F4 to enter the linux shell. Type:
cd /opt/retropie/configs/all
sudo nano retroarch.cfg
Scroll down and find these 2 lines:
video_fullscreen_x = "0"
video_fullscreen_y = "0"
Change them to:
video_fullscreen_x = "1600"
video_fullscreen_y = "1200"
Hit Ctrl-X, and Y to save the file.
Everything should work now just as well as it did before you replaced your monitor!
