I agree, money orders are good because they are paid in cash up front. Perhaps, look into what money orders look like and how to detect fraudulent ones. With official money orders, you don't have to wait for the check to clear. Take a copy of each one before depositing because the tenant won't have as good of a copy trail like checks offer. And you need permanent records My concern: why does she prefer money orders? Some potential answers: 1)She's not good at balancing check book/ credit obligations 2)illegal income - not on taxes, etc (this could be from working under the table to drug deals to "tips" at a "gentleman's" club, who knows and for some of it, does it really matter as long as she pays on time and the money order is good and other lease obligations are met.
But not to get too worried about, I think money orders are very acceptable.