The organic chickens and cows you raise don't have to come from a long line of organic livestock. There are varying requirements for the origin of organic livestock, depending on the type of livestock.
National Organic Program standards; § 205.236 origin of livestock, state that in most cases, any livestock products that are to be sold, labeled, or represented as USDA certified organic must come from livestock that is under continuous organic management from the last third of gestation or hatching. That said there are some exceptions to this standard.
Poultry exception: Edible organic poultry products must be under continuous organic management starting no later than the second day of life.
Dairy animal exception: Dairy products, including plain old milk, must come from animals that have been under continuous organic management for one year prior to the production of the milk. However, dairy animals may eat crops and forage from land that is included in a dairy farm's organic system plan (OSP) and still in the third year of organic management. Lastly, once an entire herd has been converted to organic production, all dairy animals must be under organic management from the last third of gestation.
Breeder stock exception: Right now, a semi-controversial exception is that any livestock used as breeder stock may be brought from a non-organic operation. However, if the livestock are gestating and the offspring are to be raised as organic livestock, the breeder stock must be brought to the organic facility no later than the last third of gestation.
This doesn't work the other way around though. Organic animals cannot be removed from an organic operation and placed in a non-organic operation and still be sold or labeled as USDA organic.
Keep in mind: as with any organic operation, detailed records regarding organic livestock origin must be kept up-to-date and accurate.