Does God allow polygamy today?

The original design given by God for a marriage, predates any man in the Bible that had multiple wives (Jacob, David, Solomon, etc.). Just because God allowed these men to have multiple wives, does not mean that He approved it. One could assume that God allowed these men to have multiple wives in order to show grace upon the women of this time. Moreover, Genesis 1 and 2 are clear as to what God had ordained as a marriage institution then and today:

GEN. 1:27 So God created man in His own image, in the image of God He created him; male and female He created them.
GEN. 1:28 God blessed them; and God said to them, “Be fruitful and multiply, and fill the earth, and subdue it; and rule over the fish of the sea and over the birds of the sky and over every living thing that moves on the earth.” 

GEN. 2:22 Then the Lord God made a woman from the rib he had taken out of the man and he brought her to the man.
GEN. 2:23 The man said,
“This is now bone of my bones
    and flesh of my flesh;
she shall be called ‘woman,’
    for she was taken out of man.”
GEN. 2:24 That is why a man leaves his father and mother and is united to his wife, and they become one 

In both of these Genesis accounts we see the singular use of man and woman, never plural. This is a one man and one woman design from the very beginning of creation. 

DUET. 17:16 In any case, he is not to acquire many horses for himself, nor shall he make the people return to Egypt in order to acquire many horses, since the Lord has said to you, ‘You shall never again return that way.’ 
DUET. 17:17 And he shall not acquire many wives for himself, so that his heart does not turn away; nor shall he greatly increase silver and gold for himself.

Again, God is calling the Kings of this time to abandon the idea that multiple wives is God's desire among His people. 

For example, check out these few scriptures from the New Testament:

MATT. 19:5 and said, ‘For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh’?
MATT. 19:6 So they are no longer two, but one flesh. Therefore what God has joined together, let no one separate.”

EPH. 5:31 for this reason a man shall leave his father and mother and shall be joined to his wife, and the two shall become one flesh.

1TIM. 3:2 An overseer, then, must be above reproach, the husband of one wife, temperate, self-controlled, respectable, hospitable, skillful in teaching, 

1 TIM. 3:12 Deacons must be husbands of one wife, and good managers of their children and their own households. 

 TITUS 1:6 namely, if any man is beyond reproach, the husband of one wife, having children who believe, not accused of indecent behavior or rebellion.

In conclusion, the marriage union designed by God is a mere picture of God's relationship with His people. He sacrificed Himself, in the Person of Jesus Christ, to save us from wrath. Marriage is a picture of something far greater than worldly pleasure or submission, it embodies how Christ (Bridegroom) sacrificially laid His life down for His Church (Bride).