when i first started to commute it was my hands that hurt most! not only from the vibration but i would cycle with the hands on my hoods and i had strain between my thumb and forefinger because of it. every time i picked up something heavy i could feel the strain for ages!
one of the things you can do is get some padding under your bar tape. I had this last time i had my bar tape on. (the bar tape and padding cost me ten quid together), or change the bar tape to the tape that Specialized use on the Roubaix bike (ten quid on its own). i've just changed my bar tape to this. the shop said, when i was asking for it on the phone, they wouldn't put the padding underneath as it has a padded backing and it would be too much (the bar tape looks like terry towelling and it is padded out nicely, i'm not sure if the shop kept the padding underneath or not in the end but if they didn't then its definitely sufficient on its own).
Or the other thing you can do is get some padded gloves. they cost about a tenner too. when trying them on make sure the pads sit nicely when you want them.
Together, both these things should reduce any vibration sufficiently to it is more comfortable for you, and soon you won't even notice it anyway, once your hands toughen up a bit.