You should look at the threads about Visa/MC/Diners and AX.

The main problems with non airline credit cards:
1)Your free ticket is capped at a certain $amount---if your ticket is above that amount, you pay the difference. This could be very costly if you want to travel at the last minute or do not have a Saturday night stay.
2)You don't always get to travel on the airline of your choice.
3)Any changes to your award travel tickets are governed by the fare rules, not FF mile rules---this would be very costly to do.