[sepia]

I usually just create a new blank layer on my layers pallete, select the piece I want on the original, put a very small feather on it, copy and paste onto the blank layer, then throw out the original layer by dragging to the trash. Then, save the part you want to use as a .tiff. TIFF takes up more space, but if I choose to go back and play with it more later, I can because the layers are still intqact. Also, mine always save with no background. I believe psd does that also, but you lose your history after first opening if you use that.
But, I also do things in a very complicated way. I find the very slight feather makes it look a bit less "cut out".