Just out of curiosity, why go out of your way to put straw into compost? Originally compost was the end product of the decay process on materials you had laying around. What might have originally been waste materials could become useful again by allowing them to decay. So things like animal manure, dead or "retired" plants, lawn clippings, kitchen scraps, etc. would go into compost. In the times since Rodale first converted people from manure to compost, in the 30s and 40s, composting has become quite a science. But it does not have to be. It is the same process. Put in whatever used to be alive and it will decay/compost into a nice, aromatic, and valuable soil amendment.
The process of composting almost always turns chemicals into organic materials. Picloram and clopyralid are two herbicides which survive the compost process. Thus grasses which have had those materials sprayed on them should not be mowed and put into compost. Furthermore those materials survive the digestion process when fed as feed to cattle and should not be used in compost.
An excellent substitute for straw is leaves. Watch the neighborhoods in autumn as the leaves drop. You will find hundreds of bags of fallen leaves sitting out on the street just waiting to be picked up by YOU for your compost pile. Trees are almost never sprayed with chemicals, so they make a good source of stuff for your compost pile.
You might want to examine your organic goals. There are as many different flavors of organic gardener as there are vegetarians. Some of my lawn friends are as organic as organic can be until that last application of fertilizer at Thanksgiving. Then they go for the blast of fast release nitrogen only available in chemical ferts. If you are going for official certification of your garden as organic, then you must adhere to organic materials. If not, then there is no organic police going to come knocking on your door.
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