A lot of motherboards used cheap capacitors in the power section, usually near the CPU. If they are bulging or have crusty stuff on the tops, there's a good chance they are the culprits.
It's going to be tough to troubleshoot without extra parts. But every main board I have seen has had some diagnostic tones, even with no memory or graphics card installed. Pull out everything, make sure the internal speaker is properly connected, power it up and listen for beeps. No beeps usually means a bad motherboard, cpu or power supply. You can check the supply with a multi-meter, but that will be the only thing you can test without parts to swap.