You can take hardwood (leafless) cuttings in the fall, winter, or early spring. Softwood (new leaves) cuttings in spring, or semi-hardwood cuttings (leafy, but with some wood in the stems) throughout the summer. All three methods are fairly easy with mulberry, since it's one of the easier fruits to root from cuttings. If you don't have a mist system and don't want to be bothered with tents, jars, or other methods of keeping leaves moist, I'd recommend winter hardwood cuttings. Stick more of them than you want new plants, since you'll not get 100% success, but some of them should go. If you do have some method of keeping leaves from wilting, semi-hardwood cuttings can give nearly 100% success in summer. The old-fashioned way of doing that is to stick the cutting in partial shade, and cover it with an empty mayonnaise jar, or some other large, clear container, or a large zip-lock bag works well, as a tent to keep some humidity in during rooting. Don't put it in full sun though, or it will get too hot inside. Of course if you do have access to a mist system, that's best, and will give nearly 100% success with mulberries.