My old motherboard recently died (atleast, I think it did), and I bought a new one. I just set it all up, but when I turn it on, the monitor doesn't pick up any signal, and I can't turn it off without using the power supply switch. Everything is running inside, all the lights work, etc. I removed everything except the motherboard and the CPU, and I get the same problem. Is it the CPU thats broken, then?
First, if you hold the power button for a couple of seconds it might power off the system.
The CPU could be broken, of course, but in my experience, it is very uncommon as compared to other parts, the PSU in particular, which produces the same symptoms and is present even in your minimal test. That would be my first test, based on probability.