I have the same problem, here in Virginia.

I cut them back and apply Roundup, very carefully to the stump.

Keep a few for the delicious berries!

It will be a constant maintenance problem, as birds do seem to spread the seeds with their droppings. :(

You've probably got the same non-native, invasive species as I... Himalayan Blackberry. Below is a link to help you determine if it's your species.

Please DO NOT use Round-up in wetland areas; it's can adversely affect certain aquatic animals especially amphibians. There are supposedly other glyphosate (the main active ingredient in Round-up) products that are relatively safe to use in these situations, but they would need to be used very carefully...

Here's an additional link for safe glyphosate use: http://www.for-wild.org/download/roundupmyth.pdf

Here is a link that might be useful: Himalayan Blackberry

I've got some wetlands, stream areas on my property as well and have been gradually taking out the Himalayan Blackberry. It is an invasive non native plant, although it sure is delicious. I don't know how big a property you have to clear, but I have just made it a steady clearing project every year until we have very few of those plants left. Pull up the roots as much as your body will stand and then tackle it again later and you'll make progress. I find that the native Salmonberry will fill in nicely. Not quite as yummy, but still a nice snack! Be careful of any chemicals you use around that wetland.

Goats eat Blackberries