I know that the answer is yes, I should, but outlets near the setup are not grounded (even though they look like they are) and I don’t want to have wires running though my living room.
The real question is what are potential problems ? Occasional system reboots? Permanent damage to PSU? Permanent damage to other components?
$10-15 will get you an outlet tester at just about every home improvement store. You plug it in and the three big LEDs light up and you compare it with the sticker on the device. Get one with a GFCI tester built in, when you press the button it will short to ground and if your receptacle has GFCI protection or is on a GFCI protected circuit should trip the GFCI protection.