While your cuttings take root, keep them covered and moist. In a garden bed, a simple DIY mini greenhouse does the trick. Just place a bell jar, a garden cloche or an overturned mason jar over the cutting. A clear plastic bottle with the bottom cut out and the cap removed works, too. Water the soil regularly to keep it moist, but not soggy. Your mini hothouse will keep the humidity high inside.
Once you have had several weeks of below-freezing temperatures, cover the base of the rose with 12 inches of soil or mulch, and then cover the canes with straw, leaves, pine boughs or even foam insulation. Climbing roses can be wrapped right on their supports, or you can lay them on the ground and cover the canes with straw or brush. In severely cold climates, hybrid teas are sometimes partially dug up, laid down onto the soil, and the entire plant is then covered with more soil or mulch.
There will be disappointments along the way, when many rose seeds fail to germinate, or perhaps they do germinate, only to die several weeks later from damp-off disease. Sometimes a rose seedling will turn out to be as ugly as sin, or very sickly, and you will reluctantly have to throw it away. Other times they will be just ho-hum, or look too much like its parent, and therefore have no value. But, just when you don't expect it, you might discover that one of your rose seedlings turns out to be very special.