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As the grain is reaped from the plant stalks that become the straw some of the seeds may not get picked off and can stick to the head of the straw, the hawn. When this straw is put down as a mulch these seeds can germinate. Many people think these are "weed" seeds when they are simply more of what was on the straw, the Oats, Wheat, Rye, Rape, etc. that was planted and they are not really a problem.

The only thing that is called straw that does not have seeds will be pine straw which is dried pine needles.

I would have loved to see the look on the farmers faces when you were talking to them. LOL

Aside from the pine straw maifleur mentioned, you won't be able to get agricultural straw without any seeds because of the definition of straw.

What you want is hay, which is cut before the plants have gone to seed and matured. Then the only weed seeds you have to worry about are the early ripening ones, like dandelion, that may be growing in the field if the stand is a little thin. It will cost more than straw because it has a much higher feed value than straw does.