What kind of dizziness its it, a faintness or more like things are spinning which is more vertigo?
Have you had any other symptoms?
I'd go see your GP for a general check up, blood pressure, blood count, blood sugar, ears etc...
How is your diet in the mornings? Do you eat or skip breakfast?
Don't scare yourself with something like Meniere's yet (no offence intended Ethel) as it's best to rule out simple explanations first.
I have menieres (as does, I think, Ethel) but the synptoms of everyone I've known who has it have been a lot more severe ie being thrown to the floor in vertigo attacks, hearing loss, tinnitus etc...