As long as you put thin plywood or panelling or something else peckproof over it, it's probably the best insulation material you can use, in terms of performance, performance when some dampness gets into the walls, and resistance (NOT immunity) to mice taking up residence.
You do have to cover it with something peckproof tho.
Straw is a good insulator, the down side is that it is also great bedding for rodents and for ticks, mites, lice...foam or fibreglass is worth the investment and stays put, though no insulation is rodent-proof.
You would be far better off with a non-organic material. Straw stuffed in the walls will create mold, rot, and a mouse farm. Also it does not really have very good R value for just that thickness and settles pretty quickly creating uninsulated spots near the top.
If you really, really wanted to use an organic material, wood shavings are far better than straw; if you stuff them into plastic bags you will not have problems with mold or with dampness decreasing the insulation value. Mice and settling are still a problem.
It would really be a lot better to scrounge some real insulation though. Even just styrofoam sheets (ask around at a bunch of stores, eventually you will find one with styrofoam that merchandise came packed in that they don't want to pay to put in the dumpster). Works a lot better, less chance of damage to coop or chickens.
The reason the leaves mentioned by the o.p. are not such a problem btw is that they would be in plastic bags and stacked around the outside of the coop -- that is different than putting htem inside walls.