For root cuttings, Jan-February is when you want to dig and cut. Store in moist paper towels in a bag in the fridge. Plant 4 - 6 inch long pieces in Mar-April, 2 inches or so deep and voila. This works very well for wild blackberries and tame but doesnt work for trailing type cultivars.
For leafy stem cuttings, cut a couple inches of a leafy tip in mid-Summer, add rooting hormone, place in moist ground, put a mason jar over it, and it will root if we have enough rain or if you can water it. The mason jar helps keep some moisture in though and acts as a minigreenhouse so you will have decent sucess with this method during most summers.
Just a few of the many things that my great grandmother has taught me, she is a walking book of knowledge on propagating nearly everything that produces food. Granny doesnt use rooting hormone though, thats something I added because I have better luck with it.