Can I deduct medical board exams and review course costs (ex. ABIM, ASCeXAM)?

Unfortunately no.

If this is accreditation exam that is not required by your employer but is an initial exam, the IRS will not allow you to claim a miscellaneous deduction.

According to the IRS, when the expense involves Professional Accreditation Fees, you can't deduct professional accreditation fees such as the following: 

• Accounting certificate fees paid for the initial right to practice accounting.
• Bar exam fees and incidental expenses in securing initial admission to the bar.
• Medical and dental license fees paid to get initial licensing.  

For more information about what is allowed and what is not allowed as a miscellaneous deduction, please refer to the following link:

https://www.irs.gov/publications/p529/ar02.html#en_US_2015_publink100027031

I think this answer requires more input. Some Board exams are required for licensing (USMLE Step 1, 2 and 3). However further specialty boards (Surgery, Cardiology, etc...) are not required for medical licensing. Therefore they would not fall under this definition. Can anyone clarify because I was under the understanding that specialty boards are deductible.

Despite the 2019 date on the original question; the above is out of date (2019 was the transfer date from the old forum). 

The tax law changed in 2018.  Job expenses are no longer deductible and that includes license fees.

 Even in the "old days" , job expenses were only a misc. itemized deduction.  You only got to deduct that portion of  your misc. itemized deductions that exceeded 2% of your AGI,  and then only if your total  itemized deductions exceeded the standard deduction (which was doubled under the 2018 tax law. .  (2% rule explained: https://ttlc.intuit.com/questions/2902781-what-is-the-2-rule ).

Deductible as what? 

There are no W-2 job related deductions and it doesn't sound like education expense.