In the process of troubleshooting, I noticed that I didn't have an internal speaker plugged into the motherboard which meant that if the BIOS was giving me error beeps, I wasn't hearing them. I also noticed that my case doesn't seem to have an internal speaker. So I cannibalized an old empty ATX case I had sitting around and removed its little speaker. I plugged it in and powered on. I was greeted with a long continuous beep.
I did some googling and decided that the long continuous beep may indicate a RAM problem. So I swapped out the RAM (which I had not tried yet I am ashamed to admit) and the machine POST'd! Woohoo!
I now have to plug in all the stuff I unplugged and see if I am really back in business or not.