I'm admittedly having trouble finding a place that does them, but Wikipedia claims that debit is also done through UnionPay, American Express, Discover, STAR, JCB, and PulseDo they exist? I don't know where you are, but all the high street banks here in the UK use those. Maybe some of those posh banks which charge you to bank with them have some other way of doing it, but I'm not sure the pay terminals in the shops and ATMs will work if you don't have a Visa or Mastercard logo on your card. And Amex isn't taken anywhere.
They used to have an alternative here called 'Switch' (debit only) but I haven't seen that in years. Maybe it was phased out.
EDIT:
It looks like I might be able to get a discover debit card with Capital One