The land bridge occurred during a time of significant glaciation - this means that both the Asian and North American ends of the land bridge were largely covered in ice sheets, which are relatively inhospitable to most animals and to people.
I think it's unlikely that much exchange would have taken place, even had the proper conditions extended until relativity recently, because few creatures (human or otherwise) would have survived the crossing.
Now, assuming you create a world with a more modern land bridge, the most important things that could cross the bridge would probably be horses and diseases.
Most potential draft animals in the Americas disappeared about the same time that human habitation first occurred. It is possible that the first human settlers hunted them to extinction. Lack of draft animals puts a limit on population size, and larger populations are associated with technological progress and economic might. Re-introducing the horse prior to the Columbian Exchange could have HUGE impacts.
This exchange could potentially have brought small pox, plague, and other diseases that were utterly alien to the New World. Since these illnesses killed off a estimated 90% of the Native Americans, introducing them a few hundred years earlier could have had a profound effect - the Native Americans could have repopulated, and would have likely developed some immunity to these diseases.
If a small group of nomadic herders bearing small pox and horses had crossed from Asia to the Americas around 1000 AD they could have significantly altered the course of human history - which could make for an interesting story!
Check out the Wiki entry on the Columbian Exchange for more thoughts on what they could bring: