I hand wash my cross-stitch embroidery unless it is done on tablecloth or clothing. I use dish wash to soak my embroidery in slightly warm water and I watch closely the process. So far no colors were bleeding. If I saw that colors were bleeding I would immediately rinse the embroidery with large amount of water.
I wash tablecloths in the wash machine on the gentle cycle. Fist I put them into a laundry bag to prevent possible damage.