According to jno`s link, the counties that don`t have cities are :-
Bedfordshire, Berkshire, Buckinghamshire, Dorset, IOW, Surrey, Northants, Rutland, Shropshire, Rutland, Suffolk, Warwickshire, Mid Glamorgan, Northumberland, Powys
The list would have been much smaller pre 1974 when metropolitan counties were introduced (another of Heath's brilliant ideas). Northumberland was without a city as Newcastle, which is geographically in Northumberland, was known as the City and County of Newcastle Upon Tyne, a county in its own right.