How Do I Fix the Red Ring of Death in the Xbox 360?

I replaced the power supply with the appropriate one (there's three apparently), and though the disk door opens, the system does nothing else, save blink red on the power gate. I have heard this can be fixed. Does anyone know how I can do this myself?

Suggestion:

There are a few ways. You can send it into Microsoft if your warranty is still good, but it seems you don't want to do that. There are a few other ways. There are guides you can get, they aren't very expensive, around $20 – $30. I heard that www.redringtogreen.com is a very good guide for the price. I don't know if it helps, but there you go.