Here is one reasonably reputable article on the subject: Aluminium & Alzheimer's, ABC Science
The short answer, as far as I can find out, it no, they aren't unhealthy. The article I've referenced gives some history as to where the belief originated.
Here's a quotation of the most relevant part of the article:
Aluminium has had bad press for a long time, mostly beginning around
the 1920s. Rudolph Valentino's death in 1926 at the tender age of 31
was blamed on aluminium poisoning from aluminium cookware - but he was
actually killed by a perforating stomach ulcer. Howard J. Force, a
self-proclaimed "chemist" added momentum to the anti-aluminium
movement with pamphlets such as Poisons Formed by Aluminum Cooking
Utensils. It was probably not a coincidence that he also sold cookware
- stainless steel cookware.
The first scientific "evidence" about aluminium's toxicity appeared in
the mid-1970s. People with Alzheimer's Disease have typical changes in
the brain that can be seen only with a microscope. They're called
"neuro-fibrillary tangles". Various studies found high concentrations
of aluminium at autopsy in the brains of people suffering with
Alzheimer's Disease - and almost always in the characteristic
neuro-fibrillary tangles in the nerves. So, did the aluminium cause
Alzheimer's Disease? No. It eventually turned out that the
neuro-fibrillary tangles were very "sticky" - and absorbed the
aluminium out of the water used to wash them.
As a further references 'Aluminium accumulation in relation to senile plaque and neurofibrillary tangle formation in the brains of patients with renal failure' states:
These data suggest that it is unlikely that aluminium plays any major
role in neurofibrillary tangle formation and that its putative role in
senile plaque formation is likely to be only part of a complex cascade
of changes.
