His name in Herodotos' Greek is Διηνέκης. In modern Greek this would be pronounced thee-ee-NEH-kees (with a voiced th at the start), but the pronunciation has changed a lot.

The dialects make a bit of a difference too -- Herodotos wrote in a variety of ancient Greek called East Ionic, but Dienekes himself was Spartan and so spoke a variety of Doric known as Lakonian. As a result he would have spelled his own name Διανέκης, with a long α -- the name's actually a word meaning "regular, without ceasing". (This difference in dialects is also the reason why Herodotos writes the Doric name Leōnidās as Leōnidēs.)

In antiquity the following differences apply:

• the letter delta was pronounced /d/
• eta (in the places where it occurs in Doric) was as a long open-mid vowel
• sigma was always a soft /s/ unless it was followed by a voiced consonant (which it isn't here)
• at the time of the Persian Wars, most Greek dialects had an accent that modified, not the stress, but the pitch of particular syllables (as in Norwegian and several East Asian languages); an "acute" accent meant that a syllable was pronounced at a significantly higher pitch than normal tone -- say a perfect fifth or so higher.

Bearing all this in mind, the pronunciation of Διανέκης would be /dee-āh-ne-kēhss/, where the /ne/ is about a perfect fifth higher than the other syllables, and the /āh/ and the /kēh/ have about twice the length of the other syllables. The vowels in /dee-āh/ should be self-explanatory. In the third syllable the /eh/ is a close-mid vowel, as in English play (in most dialects, anyway), and in the fourth syllable the /ēh/ is an open-mid vowel, as in bed (in most American accents or in British RP).

The syllable lengths and pitches could conveniently be represented with musical notation: quaver, crotchet, high pitched quaver, normal pitched crotchet (or if you're American: eighth, fourth, high pitched eighth, normal pitched fourth). There's no particular stress accent in Greek of that period, so you don't need to worry about where the "beat" goes.

Edit: stuff.