I would say italicize Greek as one would French etc.
Many letters of the Greek alphabet look like Latin letters although they do not correspond.
So remove the possibility that nu will be read as /v, or rho as /p, and so on.
Also, scholars will appreciate that this degree of contrast between upright and slanted forms is old-school (when the default Greek fonts had a slanted lower case).
Furthermore, this would give Greek the same typographic treatment as French etc., so would be consistent in an English document that quoted both French and Greek extracts.

And especially for that last reason, I would say italicize Cyrillic also.

Italicize as seldom as possible, especially when it's a different script, especially especially when it's a script that has no -or a problematic- Italic tradition.

hhp

As much as i agree with Nick’s opinion i don’t see the necessity to italicize a different script. The reason why it is better to italicize different languages within the Latin alphabet, is to indicate clearly to the reader that there is a loan word or citation from a different language following. A different script is enough visible change.
If, however one had to set several words in different languages and scripts within the same text, i would italicize also the other scripts.

James, do you by any chance have a piece of text at hand which we could use to make testings about your interesting issue?

Try this:

The Waste Land

by T.S. Eliot

"Nam Sibyllam quidem Cumis ego ipse oculis meis vidi
in ampulla pendere, et cum illi pueri dicerent: Σιβυλλα
τι θελεις; respondebat illa: αποθανειν θελω."

For Ezra Pound
il miglior fabbro.

I. The Burial of the Dead

April is the cruellest month, breeding
Lilacs out of the dead land, mixing
Memory and desire, stirring
Dull roots with spring rain…

Here's a less lyrical one, and with Armenian, not Greek or Cyrillic:
"
However in the Armenian text all surnames are subjected to the conventions of the reformed Eastern spelling (for example Փափազյան instead of the correct Փափազեան).
"
For one thing, do you really want that ն to be in "Italics" fighting with the closing parenthesis?

hhp

Here’s a short example from some of my text.

Similar in meaning to Hebrew אָמֵן ʔāmēn and hence Greek ἀμήν amḗn and Russian ами́нь amínʼ (the Russian reflecting shift of Greek η ē to i), though not actually used in translations of religious materials.

Here’s another longer example with only Cyrillic.

The scientific names all come from Russian, and were originally given by Johann Julius Walbaum during his time in the Russian Far East. The name O. tshawytscha is from Russian чавы́ча /tɕaˈvɨtɕa/ which is itself from Itelʼmen čevičev. O. nerka is in Russian не́рка /ˈnʲerka/, but in Siberian dialects ня́рка /ˈnʲarka/, and which appears to be from Proto-Samoyedic *nʸʌrkə, compare Nganasan nʸorə. O. kisutch is Russian ки́жуч /ˈkʲiʐutɕ/ but in Siberian dialects ки́зуч /ˈkʲizutɕ/, coming from Itelʼmen kizuez; the dialectal form is the source of the scientific name. O. keta is particularly interesting: in Russian it is ке́та /ˈkʲeta/ but this is again not native; rather, the source is probably Evenki keːta ‘O. keta’ or perhaps Even qæta ‘dead salmon after spawning’ (unrelatedly, Tlingit also has xein /xeːn/ ‘spawned-out salmon’ < Pre-Tlingit *xaʰyn, cognate with Eyak xaːnih; also cf. PA *xʸaːn ‘old age’, cognate with Tlingit shaan /ʃaːn/ ‘old, elderly’ < Pre-Tlingit *šaʰn) both of which are themselves probably from Chukchi-Koryak, e.g. Chukchi qetaqet ‘O. keta’. The Russian for O. gorbuscha is simply горбу́ша /gorˈbuʂa/ derived from горб /gorb/ ‘hump’ and describing the pronounced hump that develops on the backs of males during spawning.

Of course this looks unpleasant here because Georgia doesn’t have all the appropriate characters and browsers substitute various fonts badly.

In the short example, putting the non-Latin script in upright form, and the pronunciation in italic, works very well.

James, I think both criteria are plausible. But in linguistic texts, you handle several groups of information: the contemporary form of a word, variations through time, its ancient form and pronunciation, loan originals, non-Latin roots and so on. From a practical point of view, I would use regular Greek and Cyrillic as the general rule and reserve their italic styles to specific groups (as pronunciation or an ancient form). Hence, you have one more option to distinguish things.

> … this looks unpleasant here because Georgia doesn’t have all the appropriate characters and browsers substitute various fonts badly.

Well, on my screen (Mac, Firefox) it doesn’t look all that badly.
However, lets have a look on it with another font:

Given that your text involves systematic use of Latin italics for phonetic notation, I would say do not italicise the Greek and Cyrillic. Where you have a sequence like this 'Russian ами́нь amín', I'd say it is more important to distinguish the Russian words from the italicised Latin pronunciation than to further visually distinguish the Cyrillic from the Latin.