What I do, instead of using paper or a stabilizer, is *heavily* starch the background fabric before cutting. I use a 1:1 solution of Sta-Flo laundry starch and water, paint this solution onto the yardage with a large wall painting brush, wait a couple of minutes to make sure the starch has had time to penetrate the fiber, toss in the dryer, then iron with steam. After that I cut the background squares for the applique, always cutting about an inch bigger than the pattern states in case there is any shifting (although I haven't had any using starch, even when the edges are satin stitched; it's just my safety policy). Starch stabilizes the background fabric so the bias doesn't stretch as you are sewing around the appliques.
As others have mentioned, doing machine applique without the background stabilized almost always results in some distortion of the finished square, tunneling, etc.
Edit: The starch method works equally well on washed and unwashed fabrics.