What was it about the death of Jesus that allows God to forgive us? I've heard many people state that it was the punishment inflicted by the Romans that allows God to forgive us. This just doesn't make any sense to me. 

How can a man suffer for a short while here on earth in order that a divine being have the right to forgive sin?

What is the framework of substitutionary atonement that makes Christ's death effective?
Let's start with your question title. For the sake of argument let's turn it around and ask "Why would God not be allowed for forgive us?" The problem lies in God's idea of justice. We could spend a lot of time one this one, but let's keep it it simple:


  Isaiah 61:8 (ESV)
  For I the Lord love justice; I hate robbery and wrong [...]


In order for Him to actually be just, he has to administer justice. In order for justice to be adminstered the proper recompense has to be made for any wrong-doing. We know that the only way to pay for sin is death:


  Romans 6:23 (ESV)
  For the wages of sin is death [...]


So we have to die. Or somebody has to die for us. The only person qualified to do that is Christ. The why is a subject for another question. Let's skip to what Christ did. When he was on the Cross he took upon himself the full wrath of God against all our sins.


  Isaiah 53:4-5 (ESV) 
  Surely he has borne our griefs and carried our sorrows; yet we esteemed him stricken, smitten by God, and afflicted. But he was pierced for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his wounds we are healed.


God chose to crush Him. The pain inflicted by the Romans was nothing compared to the weight of that wrath. Even humans can endure the torture of crucifixion for the amount of time Christ did ... longer in fact. The difference was that while the physical act was in progress he also bore an infinite amount of punishment from His own Father. How does that work out for those who believe in Him?


  Romans 5:9 (ESV)
  Since, therefore, we have now been justified by his blood, much more shall we be saved by him from the wrath of God.


To round this out I would point out that from a Reformed Protestant perspective, the notion that anything the hands of men could inflict upon Christ, no matter how brutal the Romans were could do anything to change God's ability to do or not do something is considered heretical. What Christ endured at the hands of the Romans an in crucifixion is both symbolic of a spiritual reality that was going on behinds the scenes, but also real in that that it put the rulers and authorities of this world to open shame while at the same time setting the stage for his physical resurrection to demonstrate his victory over death.


  Colossians 2:14-15 (ESV)
  [God] by canceling the record of debt that stood against us with its legal demands. This he set aside, nailing it to the cross. He disarmed the rulers and authorities and put them to open shame, by triumphing over them in him.