If you don't trust your new tenant, why did you rent to them in the first place?
Is it because you live out of town or out of state that you require the checks ahead of time?
In several states the law provides for fines and or imprisonment, but does not specify if the crime is misdemeanor or felony.
Sorry, you'll just have to accept them each month like everyone else.
They can always close the bank account that the "post dated" checks were drawn on and you'd still be behind the eight ball regardless.
In some states there is a criminal offense only when the bad check is given in exchange for property or for a present consideration.
In other states it is a criminal offense to issue a bad check with intent to defraud or with knowledge of insufficient funds.