I hate to tell you but the lines probably will not wash out. The older kits were printed by setting the designs with heat.
They fade over time but usually never disappear completely and this is with multiple washings.
You could try by using a Q-Tip and a bit of water to try a small area that you can embroidery over. Is there printing along the edge of the blocks that you can try to remove. There was usualy a design number or company printed along the edge. If so you could try various detergents to remove.