I've made a lot of cross stitch ornaments over the years!
If on perforated paper or plastic canvas, I just cut them out, then back with either felt or pretty fabric or scrapbook paper glued together with acid free fabric glue.
If on fabric, I have cut them out, cut out a shape in cardboard (use a PH pen from the scrapbook section of the craft store to make sure it's acid free!) and sort of "wrapped" them around the cardboard, gluing in place, then backing as above. Sometimes I added a bit of lace or other trim around the edge.
You can also sew them like a pillow with a fabric backing and stuff them with polyfil for a more 3D effect.
I usually add a loop of ribbon or cording for a hanger as well. If gluing on a baking, I just make the loop and glue it into place before adding the backing. If making a pillow, I sew it on the spot I turned the pillow, which hides the turning stitches as well.