Most 3D cross stitched projects that I have seen have a line of back stitching at the seam edges. When you finish stitching all of the pieces you crease the fabric under at the back stitched lines and then stitch the pieces together. The thread that you sew with goes, not through the fabric, but rather under the back stitches. Here is an example:
Instead of using a back stitch I sew my ornaments together using a ladder stitch. Click here to see a youtube video of a ladder stitched seam. It is not difficult but I think that it may be a bit slower. And I find that the last seam tricky.
For my ornaments I stitch around where the seam will be using a temporary running stitch with Sulky Sliver Metallic thread. The running stitch for the sloped sides on my ornaments goes up two threads and over one thread.
Jean Farish stuffs her 3D ornaments with fibrefill. I use card stock to stiffen the fabric pieces instead.
TO STIFFEN WITH CARD STOCK
Here is a pattern of diagram for the card stock shapes that I use. It is shown as if drawn on 1/4 inch graph paper.
I used a blunt yarn needle, or something similar, to score a line on the card stock between the center rectangle and the two triangles. Then I fold gently along that line. To attach the card stock to the stitched pieces I put spots of fabric safe glue on the back of the card stock and fold the seam allowance over. The glue that I used is FabriTac. I don't like it for everything, but for this project it is nice that it sticks so quickly.
I use fray reducer and I only cut the seam allowance just before I am ready to stitch two pieces together. And then I handle them as little as possible. So if I were writing instructions for someone else they would go something like this:
MY INSTRUCTIONS FOR FINISHING 3D ORNAMENTS
Using the diagram cut 5 shapes from stiff card stock. Cut one of the six sided fabric pieces leaving about 1/2" seam allowance outside the running stitched line. Finger press along the running stitched lines. Lay the fabric shape right side down and lay a card stock shape on top of it. (Since you have folded the card stock already the triangles will point up towards you a little.) The running stitch lines on the fabric should just show around the pattern. (If the card stock piece is too small you will want to cut a new one out just a little bigger.)
I would not try to glue down every bit of your folded over fabric. There is too much fabric at the corners and you don't need it all glued down just to keep the card stock in place. The glue sets very quickly so I would just put a spot of glue near the top and bottom points of the card stock and then fold the fabric points over. Hold them for a few seconds. Then put a spot of glue on the two little short sides, fold the fabric over and hold the two spots for few seconds. Repeat for the other 4 stitched pieces. By then the first two may be dry enough for you to sew them together.
When I ladder stitch two pieces together I start at the top and go to the bottom. My needle follows the path of the running stitches. I remove the running stitches when I have finished ladder stitching the seam. Before I sew the last seam I add a loop of string to pull the tassel and hanger. (See below) For the last seam I use very long pieces of thread because I keep the stitches very loose. (I keep the stitches loose because for some reason I need to be able to get at least one finger inside when I am doing the ladder stitch. It may be a personal quirk.) I stitch from top to middle and then use another piece of thread to stitch from bottom to middle. Then I carefully pull the stitches tighter. Here is a (squished) picture of the ladder stitches on the last seam.
ADDING THE TASSEL AND HANGER
I wish that I had thought to take some step by step picture of this process but I hope that these illustrations will be an adequate explanation.
I make the tassels for these ornaments before I sew the last seam. I make the string on the top of the tassel long enough to pull up through the ornament and become the hanger. About 16 to 18 inches of thread folds to about the right length.
One could insert the string for the tassel before sewing the last seam, but it is a little awkward sewing the seam with the tassel flopping around. Instead, before I sew the last seam, I insert a fold of strong thread that I will use to pull the tassel. To do this I take a about 18 inches of strong thread (something like DMC pearl #12) and fold it in half to make a loop. Then I insert it inside the ornament with the loop sticking out the bottom and the ends sticking out the top. To keep it out of the way while I am stitching the last seam I tie the ends through the loop. (This strong thread is illustrated in pink.)
When I am finished sewing the seam I put the long threads of the tassel through the loop at the bottom of the ornament. (Though it is not well shown on the diagram the length of the string between the tassel and the loop (A) should be a little longer than the height of the ornament (B).)
Then I pulled on the strong thread at the top to pull the tassel snug against the bottom of the ornament.
I added my tatted bead cap and a small pearl bead and then made a small knot. Finally I tied a knot a few inches further up the string and cut off the excess.