You'd want to take a non-flowering branch. It will not root in water, use a special soil-less potting mix and rooting hormone, and put it in a pot in shade under a plastic bag. Keep it moist and *shaded*, but ventilate it if it looks at all soggy. In a few weeks you will see some new leaves emerge, that means the cutting has rooted.

These root "fairly easily" but the trick is to get them to survive their first winter. There's another thread on this forum where ego gives great instructions on rooting hydrangeas and overwintering the cuttings. (you can search the forum for it). He suggests taking the cuttings in very early spring to improve survival rates. I've had some cuttings overwinter, but I've lost a lot of them too.

Sometimes you can inspect the area around the root ball and see a section that can be removed - I've done that too with one particular favorite plant. This is probably not the best time to do it, though, since you don't want to take a chance and disturb the roots on a flowering shrub in its prime flowering time.

Layering also works well for hydrangeas. Mine seem to do that all by themselves each season. Just take a non-flowering stem that's close to the ground, pin it down with a rock, & check it next spring. I don't even nick the stem where it meets the soil. I've also broken stems by accident & tossed them behind the shrub out of sheer laziness & found a rooted plant come the next April. Overall, I think they're pretty easy to root...I wouldn't "over think" it. The layering method also eliminates the over wintering issue since the new plant is still attached to the mother shrub.

Note: Our soil is very high in organic matter. We mulch every year with 6" of 50/50 leaf moulde & composted manure. Most things root easily in our soil...some things, too much so!

Tricia

If you have unflowering branches, cut them in a such way that you'll have at least 3 pair of buds (4 is better) on a lenghts of the stem.
Cut off the leaves from the two lower buds and leave the top pair with the leaves (you may cut them in a half to reduce transpiration).
That's it. Insert cutting in a pot making sure that both lower pairs of buds are in the soil and top one is above the ground.
That's it. Water well at once. Then follow Nan's instructions as above.
Good luck

Tricia is super-right, ground layering should be a method of choice if you want to propagate your own shrubs.
Eliminating overwintering process will give you much better rate of success and much stronger/healthier plants next spring.

Another strong vote for Tricia's recommendation of layering with a rock. I find this almost foolproof compared to cuttings. You also tend to get a more significant starter plant this way with no extra care until you dig it up. I have started to do this with my clematis as well this season so I am hoping for similar results.