Is it legal in NH to be charged first and last months rent plus a security deposit?

Yes it is legal to ask for first, last and security in Rhode Island.

Page 5 of The Rhode Island Landlord-Tenant Handbook issued by the State, states that, "A landlord can take a security deposit from a tenant at the beginning of a new rental term but it cannot exceed one month's rent." and "Separate amounts of money can be requested from a new tenant for prepaid rents, etc." You can view the handbook (right click to download the pdf) on the link below.

We will be doing this from now on after a long-term tenant used his security deposit to pay his last month's rent. While the apartment was fortunately in good shape, it could have been an expensive lesson if he'd trashed it and skipped out on the last month's rent. Collecting last month's protects the security deposit and ensures that the rent will be paid once they've notified you of their intentions to move out.

Since paying all three fees (first, last, security) at once can be a huge financial hit, you may want to have them pay one of the fees in installments over several months.