Several ideas I've had that have passed through my head and I was wandering if anyone has tried them or did them and what your results have been.
1. Instead of using toilet paper/paper towel for the base/top layer instead use cotton cloth, something like part of an old tshirt that you don't wear anymore. It would be reusable significantly longer than a paper towel or toilet paper.
2. Taking the plastic 'knitting' sheeting, aka the tops for shoot jars, and using that as a tray. Sew/knit them up into a tray position, a full sheet for the base and then cut another sheet to make the four sides. It has the natural drainage that you would need and combine that with anything, toilet paper/paper towel/cotton cloth you would have a capture method for holding dirt for microgreens. They would be small enough to make them quite easily transportable and they could be hooked together and with a little ingenuity could provide for a angled, irrigated setup so you didn't have to manually water.
3. During the germination process, initial watering(I don't think I would want to do it after they have started to sprout) use urine as the water source. I would think before the seed opens you wouldn't be able to hurt the seed any with urine, but once the sprout starts to emerge it would be another story altogether.
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Dave Burton's Boot Adventures at Wheaton Labs and Basecamp