I think you would need one powerful mower to harvest mature grains. I mow my orchard when the grasses are in the green seed stage, which is when I assume you would plan to harvest? I dont think any standard bag mower could handle the amount of material I cut. I use a flail mower, which works well, but would be useless for your purpose of collecting the grain as it just chops everything up and leaves it on the ground. Did you have a certain type of mower in mind? I am curious about the idea, but question its feasability. Good Luck!
Its not going to work with corn, but I was thinking it would work with lighter grains such as millet, amaranth, hulless oats . . .
The mower will pick up the stalk and a good bit of the seedhead. I'd probably just save it all - the chickens can pick through and find the seeds and the rest of the material would become litter. Run the chickens over the area after mowing, and what they do not find re-seeds for the next year.