OR--you can let them grow. Crazy, right?
I have a feeding station, more like a cafe. 6 feeders, two baths, decaying logs strewn about, and I've planted woodland sunflowers and a Jerusalem artichoke (both native and needing no extra TLC). So, when the sunflower seeds I put out aren't eaten, they grow. The thistle for the finches doesn't grow, and neither does the safflower (that I know of).
If there's filler in your feed, you will get filler weeds. The birds ignore what they don't care for, and they could care less that you paid hard earned $$ by the lb for the filler I recommend buying single seed bags, if you want a mix you can do it yourself. That way the only thing that grows under your feeder is something that will also benefit the birds.
Lemme see if I can get some pics up for you...so you can see the worst thing that could happen for youself
ok, pics are in my gallery!