I have several clematis varieties, and I'd like to root some cuttings. Any advice on when and how?
From an extension service publication:
Homeowners may have success propagating clematis by cuttings or layering. All types can be increased by cuttings taken in May or June from half-hardened shoots of the current season's growth. Use a rooting mix of two parts sand and one part peat and a rooting hormone (available at garden centers). Supply high humidity, warmth and light in order for the cuttings to root within four to five weeks. The large-flowered hybrids will take more time to root; if cuttings are taken in May, they may not root until late August. If rooted by early August, plant them out. If no rooting occurs until late August, hold plants over winter in pots and plant in early spring. Layering is the easier method and can be done in the fall. Choose a mature stem produced earlier in the season, or from the previous season's growth. Secure it into the soil at the nodes or bury a pot containing a mixture of equal parts sand and peat and secure the stem into this. Rooting occurs within about 12 months at which point the rooted sections can be detached and planted.
The concensus seems to be that of the common methods of propagation - seeding, cuttings or layering - cuttings pose the most difficulty. Layering may be the easiest for propagating an exact clone of the parent plant.
I agree. Depending on the varieties you are attempting to propagate, layering is definitely the easiest to do and will insure a carbon copy of the plant. The species clematis are notoriously difficult to propagate from cuttings.
If you are still interested in rooting cuttings, google clematis from cuttings and there will be a couple of good references to go by including one from the International Clematis Society.