Do any of the regional dialects spoken in Greece today preserve any elements from their Ancient Greek counterparts?

To start with: the default assumption in Greek historical linguistics is that the ancient dialects vanished under the Koine, and that the dialectal diversity of Modern Greek does not owe anything to the dialectal diversity of Ancient Greek.

That means that the null hypothesis is that there was no survival of Ancient Greek dialect; and methodologically, if you can prove a feature of Modern Greek dialect through modern mechanisms, that should be preferred over accounts using ancient dialect. To do so satisfies Occam’s Razor.

Let me take the silliest example I can think of of a proposed Ancient dialectal survival.

• The Aeolic for ‘name’, normally ónoma, was ónuma.
• Aeolic was spoken in Thessaly and Lesbos.
• In Modern Thessaly and Lesbos, ‘name’ is ˈonuma.
• COINCIDENCE?!!

… Why yes. Coincidence.

• In Northern Greek dialects, unstressed /o/ is regularly raised raised to /u/. For example, Standard Greek ˈanθropos ‘human’ is pronounced as ˈaθrupus.
• Thessaly and Lesbos are Northern Greek dialects
• Therefore ˈonoma was always going to be pronounced ˈonuma in Thessaly and Lesbos.
• In fact, it’s pronounced ˈonuma just about everywhere north of Corinth.

This means that not as much Ancient dialect survives Occam’s Razor as enthusiasts might like.

Tsakonian, by any sensible metric, is indeed a separate language. It also has clear survivals of Doric. But it doesn’t have clear survivals because amateurs like Michael Deffner said so. It has clear survivals because the magisterial neogrammarian Hubert Pernot ended up conceding it has Doric survivals, after three decades of scepticism. And in the process, he dispensed with a lot of faulty claims of dialect.

The only other widely known claims of ancient dialectal survival (as opposed to the odd word here and there—on which see Nikolaos Andriotis. Lexikon der Archaismen in neugriecheschen Dialekten) are:

• Pontic often has /e/ as a reflex of ancient eta, as opposed to the expected /i/. That has been claimed to be Ionic, with Ionic eta more like /æː/ than /ɛː/. I have to admit, I haven’t been convinced.
• There are Doric survivals in Southern Italy, Crete, and the Dodecanese. Those are at the level of individual words displaying /a/ corresponding to Attic eta, rather than the more systematic survivals in Tsakonian. It’s hard to read anything about Southern Italian Greek, for example, without seeing the word nasiða ‘strip of farmland’ corresponding to Standard Greek nisiða ‘islet’.