http://www.uarsrelief.com/sleeppositions.html
I tried the falcon position for a few days, but I prefer having a pillow under my head. I always slept with a pillow pre-CPAP and just find it more comfortable to continue using a pillow. I also found sleeping without a pillow seemed to put too much pressure on the side of my nasal mask.
I tried using the falcon with a pillow under my head, but it put to much pressure on my neck because my head was elevated above my torso; it also led to shoulder discomfort for me.
I have now found a way to make the falcon work for me and to still keep my pillow! I put the pillow under my head and under the shoulder "away" from my mask and face, lifting my torso up and relieving pressure on my neck and shoulder.
Modified "falcon" right position:
This requires two pillows (there's NO pillow under the head of the person in this photo, but I overlayed a rectangle over his head to show where the additional pillow would be located per my description).
1) From the original falcon - "For added support and comfort, you can place a pillow under the right half of your chest and abdomen."
2) I have my face to the upper-right edge of the pillow. The pillow is then angled at about 30 degrees, so the left edge of the pillow is under my left shoulder and left torso. (I like to have my mask hang off the side of the pillow, which is why my face is at the edge of the pillow. If you prefer to make a "pocket" in your pillow for your mask, adjust pillow accordingly.)

Modified "falcon" left position:
This requires two pillows (there's NO pillow under the head of the person in this photo, but I overlayed a rectangle over his head to show where the additional pillow would be located per my description).
1) From the original falcon - "For added support and comfort, you can place a pillow under the left half of your chest and abdomen."
2) I have my face to the upper-left edge of the pillow. The pillow is then angled at about 30 degrees, so the right edge of the pillow is under my right shoulder and left torso. (I like to have my mask hang off the side of the pillow, which is why my face is at the edge of the pillow. If you prefer to make a "pocket" in your pillow for your mask, adjust pillow accordingly.)

This is only a slight modification to the falcon and Eric Falcon and others previously documented this sleep position, but this small change has really improved my stomach/side sleep.
I hope this helps some of you stomach sleepers who weren't quite comfortable with the original falcon.
http://www.uarsrelief.com/sleeppositions.html
P.S.
Note to would be stomach sleepers - with this position and my head hanging off the edge of my pillow, my mask is to the side or pointing slightly down. I have noticed that for me, this tends to not work very well if I am congested. My nose congestion seems to get worse when I lay down and sometimes I can't seem to clear it when sleeping on my stomach or side. When I get congested, sometimes I end up rolling on my back. If it's really bad, I may need to sleep on my back propped up at an angle on a foam wedge or pillows. Before CPAP I was never a back or really even a side sleeper, but on CPAP when I'm congested sometimes I just can't seem to sleep on my stomach.