Shawn,
Look into 'no till farming' and I think you will be surprised. One of its principles is the soil structure is a living organism. Just like our body's skin is its largest organ, the soil is an organ. The soil needs to be covered and protected just as our skin does. Soil does not do well exposed to sunlight, oxygen, and rain. It degrades it quickly. Leaving the 'cloths' on the land will give you better results. Of course the grass needs to be controlled so it does not out compete your early plants. In no till the answer would be to plant an over winter crop, especially a nitrogen fixer like clover. Roll/crush this in the spring and then plant with minimal disturbance to the soil. On can also sheet mulch to kill the surface crop, but don't expose the soil. Cardboard works well.
Watch the first 15 minutes of this video and see if you don't have some insights on how to proceed.
http://www.backtoedenfilm.com
No till is not the only way to go and is only one school of thought. However, more and more evidence points to tillage as very bad for the sub soil 'herd' that exist on your land. Take care of the micro critters, and they will take care of the soil. When the soil is taken care of the crops grow well. Funny that...