Dizzypyxxy is correct in saying that it depends on the rose, but a few of the other points need clarification.
All roses, own-root or grafted, grow true from cuttings. The cutting is coming from the rose itself, not from the grafting rootstock..
Your cutting will not necessarily be weaker than the parent plant. Hybrid tea roses, grandifloras, and some floribundas require grafting to grow to their full potential. Other roses usually grow quite well on their own roots. If you're taking a cutting from a landscape shrub rose, one of the antique roses (known as OGR's -- Old Garden Roses), a polyantha, a hybrid perpetual, or virtually anything but a hybrid tea, grandiflora, or floribunda, the cutting will grow well on its own roots.
It's best to use a cutting that's about 6-8 inches long with a bud or bloom at the top. The rose's growth hormones are strongest when the rose is in bloom. Before you plant your cutting, cut off the buds or blooms and remove the leaves from the bottom two or three inches of the cutting. I usually dust the exposed leaf nodes with a powdered rooting hormone and then plunge the leafless bottom section of the cutting into a 1-gallon container filled with equal parts of potting soil and vermiculite or perlite (to make the soil more friable). Keep the soil in the container moist, but not soggy, and put the container in the shade for the first few days. After that, you can expose it to part-sun/part-shade for a few weeks and then move it into the sun. After the cutting begins to produce new leaves and to grow, you can transplant it into the ground or into a larger container.
Sometimes I skip the container and plant the cutting directly in the ground close to the parent plant, on the assumption that it's a location the cutting is certain to appreciate. If the parent plant is growing well in the location, so should the cutting.
