Some roses will do well on their own roots. Some won't. But there seems to be a move by nurseries these days to grow roses on their own roots where possible. You will find that older style roses, climbers, ramblers and miniatures do well like this. Hybrid Teas on the other hand can be too highly bred to perform well unless they are grafted onto a good rootstock.
I have a highly scientic method of discovering which ones do well on their own roots. I take some cuttings, and if they do well, I'm happy! If they don't do well,then you're no worse off.
If I have the choice, I take them once the summer is over. While the days are still warm, but the sap has started to dry up a little. So in the Southern Hemishere, that's usually mid March. Some people like to do their cuttings in mid summer, when sap flow is high. I have some success with that, but I find taking them at the end of the growing season, and leaving them in the ground over winter produces a higher strike rate.
Usually, I select year old wood - each cutting wants to be about 6" long, and the thickness of a pencil with 3-4 buds on it. A lot of people advise removing the leaves - I don't find leaving them on makes it any more likely your cuttings will die.
Here's what I do slightly differently. Firstly, I take the cuttings during the full moon. Yes, it really does make a difference.
And secondly, I use what I call "willow water". Strip some bark from willow branches (any variety will do). Now soak that bark in a tub with enough water to cover the bark for a couple of days.
The willow contains a chemical that helps with root growth. If you don't have access to willow branches, then a couple of asprin will make a good substitute. Once you have taken your rose cuttings, put them in a jar of this water, and let them sit for a day.
Depending on conditions, you may get the end of your cutting beginning to form a callouse like this within a few weeks. Normally, you won't see this, as you want to leave the cuttings undisturbed until they are ready for transplanting. However, I had some cuttings that needed to be moved, so I took the opportunity to photograph their progress 8-10 weeks after taking them from the winter prunings.
This one is much further advanced. Notice that there is strong root growth from the base of the cutting. This should do well and may even possibly produce a bloom or two this summer. If you absolutely have to transplant the cuttings during this stage of development (as I did), then take great care....these new roots are very fragile, and will break off at the stem if you aren't extremely careful.
Update: These rose cuttings went on to produce a fine crop of blooms in the following summer. They were probably taken from the red climber, Dublin Bay, (although I never did make an absolutely positive id of them) and with over 30 new plants, I had more than enough to give plenty of them away to friends.