I just had the exact problem on my 2006 MacPro 1,1.
I managed to get it to work. The symptoms where almost identical. It would attempt to start, and then just "pop" off again. I took all memory out (except for original ram), my 3 extra hard drives, even the video card, all with the same result.
I ended up taking the "insides" apart and thoroughly cleaning every part. I took the video card fan assembly off, cleaned the heatsink and cooling grid, I took the PSU fan apart, cleaned it. I took the front two fans out and cleaned them, as well as the back fan (by the memory). I also cleaned the cooling grid of the cpu. All these items had layers and layers of dust on them, almost like little blankets. While I had it apart I also replaced the CR2032 memory battery, since it is cheap, and I had read that there could be start-up issues if it was old..
After this, it started like a champ, and it was soooo quiet!!!