Cash is certainly not king here.
I only accept personal cheques, post-dated for the duration of the lease. I have even had people give me 2 years worth at once. The amount of time and hassle saved is priceless. Never had a cheque bounce, but I am confident since all my previous tenants have had credit scores above 720.
Do yourself a favor KaeJS, don't accept anyone that gives you any inkling of concern. Several prominent CMF landlords have already chipped in with their opinions. Borrow from this wealth of experience. In fact, you never see any of them start threads about their rentals, that's because we typically don't run into problems. The worst problem I have had in many years was a tenant that would not replace the O-ring in the kitchen sink faucet. By the time they called and had me there, they easily could have been trouble free. Minor annoyance compared to loud, fighting tenants, and ones that don't respect you enough to call ahead when they will miss an appointment.