In Genesis 30:37ff, how did Jacob's placement of rods of poplar, almond, and chestnut cause the sheep to give birth to speckled lambs?

The fact is: God blessed Jacob in spite of his superstitious use of the stripped rods of poplar, almond, and chestnut. Jacob's scheme probably depended upon a faulty notion that vivid visual impressions during the act of reproduction determined the traits of the offspring. He may have thought that placing the varying rods in front of mating animals would result in unusually coloured animals through some sort of hocus-pocus. God seems to have blessed Jacob in spite of his earthly scheming (Genesis 31:11-12). This fits into a broader scheme that runs throughout Jacob's life.

God deems to favor Jacob before even his birth. God decides to be Jacob's God before Jacob can do anything right or wrong. It is through Jacob that the whole world will be blessed and through him that the nation promised to Abraham will be born. Jacob's problem is that he thinks he needs to earn his good fortune; this is why he deceives Esau, leaves the land of promise, and tries to gain worldly wealth with Laban. Jacob does not trust that God is going to bless him and keep him, but thinks that he must take care of himself.

The simple truth is that God promised to be Jacob's God no matter what ridiculous things Jacob might do. God blesses him even when Jacob doubts God and strives in his own strength. We are not to attribute the flourishing of Jacob to his magic, but to the promise of God.