You don't need an oscilloscope. But what you do need is a good set of schematics, a DMM with Continuity Check mode, and lots of hours of your time. Even then, you may not find the problem. I myself did just that and eventually gave up and shipped my board to our own TechKnight a year ago:

viewtopic.php?p=195477#p195477

viewtopic.php?f=7&t=18026

It still hasn't been repaired because he hasn't come across the necessary parts.

You may be better off with a totally new board. My case is a bit different though because my board has a socketed CPU, which are more rare than the soldered ones.