As someone who has worked in worked in publishing for almost 40 years and who has run newspapers, I have to say that Podz's advice to "go talk to a competing paper" is brilliant.
That is, assuming there IS a competing paper. Every town had at least two and many had several 30 years ago. Now most have one daily.
Since this is an alternative weekly doing this, go to the daily.
Talk to the highest-up editor that you can in whatever the appropriate department is that might run/buy your pieces ... and also ask that person if they think there is a story in the alternative weekly's "stealing" your copy after you asked them to stop doing that?
Should the daily buy rights to your stuff, they will take care of the legal end of shutting down the weekly's unauthorized use. Newspapers have media attorneys on retainer. If the daily decides there is a news story in the weekly's thievery, then you get to put public pressure on the weekly, embarrass it, and probably end the thievery.
Copyright and intellectual property lawsuits are tricky. A good intellectual property lawyer will bill about $25k per month on a case and the case will drag out for 10 months to three years or more if both sides fight.
What will you recover?
Have you registered the copyright on each piece that has been used without compensation? If so, then you are entitled to punitive damages. If not, then you are entitled only to the actual monetary loss that you can establish.
How much value, reputation (monetized), future earnings, etc., can you document that you have lost as a result of this unauthorized use?
And then, not knowing more details, it may be that you don't have a case at all. Depends entirely upon the nature of the material being co-opted, what licensing language you have on the site, what agreements you have made in writing with the weekly, how they are using the material and many, many more little things.
Bottom line: I think a lawsuit would cost you much more than you would recover if you prevailed. If you were really, really lucky, the court might order the defendant to pay your legal costs. But most often these things are settled long before trial and both sides eat their own legal costs as part of the settlement.
That said, copyright infringement sucks.
And the growth of the internet and blogosphere have created a culture in which many people seem to think they can take and use anything they like that they find on the web.
For instance, I'm shocked at how many people think -- even tell others that -- Google Images is a good source of royalty-free art and photos.
It's not. It's an index of images on the web. And most are subject to copyright.
Way long enough here