This week's (unusually large) collection of claims from the popular press in need of a reference:

From the Turkish Weekly:

The best we can ask from the military is what Diogenes wanted from Alexander the Great: “Stand out of my sunlight.”

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Sean Manning glosses
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The story about Diogenes of Sinope and Alexander the Great comes from
somewhere in Diogenes Laertius, book 6, although I think one of the
Alexander historians may mention it.

From the American Chronicle:

It certainly seems like a good place to start the confirmation process of what the ancient Greek writer Metrodorus, 2,400 years ago, declared: "To assume that the Earth is the only inhabited world in infinite space is as absurd as to assert that on a vast plain only one stalk of grain will grow."

From the Denver Post:

Finally, keep in mind that "We dig our graves with our teeth," Fitzgerald says, quoting the Roman poet, Lucretius.

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Stieg Hedlund glosses
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There is certainly an Arabic proverb I’m aware of that runs, “He who eats when he is full digs his own grave with his teeth.”

From the Baltimore Sun:

The ancient Romans had a tradition, according to C. Michael Armstrong, former chief executive of AT&T: Whenever one of their engineers constructed an arch, as the capstone was hoisted into place, the engineer assumed accountability for his work in the most profound way possible - he stood under the arch.

From the Sentinel:

Ancient Romans used radishes for their medicinal value, believing they could stimulate the appetite, aid indigestion and treat melancholy.

We've talked about Tiberius and cucumbers before (in the context of greenhouses), but I don't remember the doctor's recommendation mentioned in this press release:

It is believed that the Roman Emperor Tiberius Ceasar, who ruled between 14 and 37 AD, was advised by his doctors to eat one cucumber everyday. Subsequently, arrangements were made to produce cucumbers round the year. Movable beds were placed out side one favorable sunny day and beds were moved inside during inclement weather conditions the movable beds were covered by frames glazed with transparent stone such as alabaster, mica, talc on winter days well manured soil in these beds was suggested, possibly, because of its heat producing quality.

This one comes from a third grader, but someone older must have told them this:

Many people in Rome believed that if you do not make a statue or portrait of a person, when they die, they will come back to haunt you.

From the Citizen Patriot:

Spellings added that teachers have been teaching to tests since the days of Greek philosopher Socrates.

From the Ledger comes one I recall hearing about as an undergrad or grad student, but never pursued:

Vending machines have been used to sell merchandise since the days of the ancient Greeks, when holy water was dispensed for a coin.

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Max Nelson glosses:
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The source for this is Hero, Pneum. 21.

See http://www.history.rochester.edu/steam/hero/section21.html

From the Citizen comes one which I thought we have dealt with before (but I can't find):

The game now referred to as hopscotch is believed to have originated in the ancient Roman Empire, where armies used 30-metre-long courses to improve the agility and conditioning of their armour-clad soldiers.

From the Times of India comes a claim which includes an interesting howler:

Pliny, the famous philosopher and physician of the fifteenth century, notes in his book that 'the reason why Rome has been able to develop into a powerful state is that the Romans regularly take sun-baths on their terraces.'

... and we conclude with something from the Freeholder:

The ancient Greeks too had imagined some of their mythic deities as having wings such as Athena (goddess of victory), Eros (the romancing assistant to Aphrodite - equivalent to Venus' Cupid of the Romans), and the wing-footed Hermes as also a messenger (aka Mercury).

Was there a Nike=Athena equation?