I tried doing the straw thing once and mice/ voles ate all of them! I had none at all. Also tried growing some in tubs and only ended up with a bunch of tiny things with the largest ones the size of an egg.
Our older saying was to plant them on Good Friday (or Saturday), but in the last few years I have been planting mine in late June with very good results- mostly due to having lots of summer rain compared to normal years when it is usually very dry. I can say that they come up better in warmer weather, and very quickly also, which is less of a chance that they will rot. I plant them the same depth either way. My family always planted by the moon signs, but I tend to plant them when weather or time permits. In my younger years we would hook a horse drawn plow to the riding mower and plow out the potatoes.