Weed control is the name of the game in organic farming.  We tilled under the cheat grass infested portion of our field before our first cutting.  Some of that area had to be mowed (and not harvested) due to pig weed infestation.  (But the cheat grass is now gone!!)

I think the mowing is okay, but the tilling should be done with care.  Broken soil seems to invite pig weed.

Next year, we will try using this old cultivator I have redone with some new fittings and chains.  We tried it out yesterday, but the pig weed was too well established and it didn't do very well.  If we use it in the spring when weeds first appear, it should pull some weeds out without hurting the alfalfa.  Alfalfa is a perennial with very deep roots.