We replaced the gutters over our front porch years ago, but left the back porch gutters alone. They appeared to do the job well enough, even with pine needles catching in the gutter guard.
However, the sight of brown stains behind the gutters on the fascia clued us into the less-than-perfect performance of these metal gutters. Since our house was getting repainted, this seemed to be the time to replace these old hunks of metal.
After removing the screens, I could see the big pins that hold the gutters in place. They're basically 5-6" nails, with the shaft wrapped in a sheath to reduce corrosion, I suppose.
For each pin (working in from the ends, toward the middle pin), I first tried prying away from the fascia. I couldn't be too rough, because 1) I didn't want to crack the wood, and 2) I was 10' up on a ladder.
Once I got the final pin out (allowing the VERY HEAVY gutters to crash to the ground), I got the wonderful surprise (not surprising given the brown drips): rot on the fascia. Actually, it was mostly the casing that was between the gutters and the fascia, so I just had to remove the rotten casing and replace it before installing new gutters (coming soon).