I need to write an essay about the culture of rewards in Beowulf. It seems to me the main characters are not motivated by rewards. Am I missing something?

I agree with you that the characters in Beowulf are not necessarily "motivated by rewards."  In fact, they are motivated by honor.  However, rewards are always a good bonus, and both of these ideas can be proved through evidence in the text.
First, let us take a look at honor in regards to the character of Beowulf as the Anglo-Saxon hero.  Beowulf's bravery, respect (of his king and his ancestors), and his abilities as a warrior all lend to his honor.  Beowulf is brave in the fact that he takes on the ultimate Anglo-Saxon nightmare:  Grendel.  Even further, he takes on Grendel's mother and the dragon.  These "monsters" are threatening Anglo-Saxon society; therefore, the hero must defend society against them.  When Grendel does away with other warriors in the mead hall and ravages the king's lands, Beowulf acts accordingly.  By doing so, Beowulf proves his might as a warrior. Beowulf even gives own life in order to end the life of the dragon.
In regards to your question, though, the characters are definitely rewarded greatly for this honor that they achieve due to their actions.  For example, when Beowulf answers the call to help Hrothgar and his kingdom by ripping off Grendel's arm, Hrothgar heaps rewards on Beowulf.  What follows is a great list of rewards that help to make up one of the conventions of an epic poem.  Yes, Grendel's arm is hung from the rafters; however, Beowulf also reaps monetary rewards.  Hrothgar gives Beowulf a golden banner, helmet, sword, coat of mail, and numerous horses.  Yet another example of reward comes when Beowulf kills Grendel's mother. Spying a sword on the wall of her lair, Beowulf grabs the sword and (with the help of the giants' magic) kills Grendel's mother.  When Beowulf returns, yet another feast is held and other gifts are given.   Eventually, Beowulf is rewarded with the greatest earthly honor:  the kingship.
However, the ultimate reward for Beowulf's honor is everlasting and results from his sacrifice in slaying the dragon.  Mortally wounded and helped quite a bit by Wiglaf, Beowulf kills the dragon with a small dagger (instead of his usual big sword).  Now it is Wiglaf who is rewarded as he is given Beowulf's mail shirt, helmet, necklace, and numerous rings.  Wiglaf is also given the kingship.  However, it is Beowulf who essential becomes eternally rewarded through the memory of his deeds.  At the very end, the reader sees the bravest of the Geat warriors singing Beowulf's praises.  This reward continues into the present day as students read Beowulf's story, giving Beowulf the ultimate reward:  immortality through his tales of honor.
Thus, not only are Anglo-Saxon warriors (and Beowulf, specifically) rewarded in the current life for their adherence to the code of honor but also they are rewarded in the afterlife and are remembered beyond their own lifetime for their valor.  With all of these things in mind, the reader can safely assume that Anglo-Saxon society certainly was a "culture of rewards."