Yes, there is a distinctive Y-Chromosome marker, often refered to as Aaron Chromosome because Kohen are supposedly all descended from Aaron (brother of Moses). Most researchers agree that this shared chromosome goes back about 3000 years. It's interesting that while normally Jewishness is passed matrilineally, being a Kohen is passed patrilineally. Some researchers have also found a similarity among Levites (second level of priesthood), with shared chromosomes going back 2000 years, but it's not nearly as clear cut as with Aaron Chromosome.
Edited to add: natrurally, as with any patrilineal issues, not all Kohens have this chromosome, but many do.