There was no single calendar in ancient times; each location had its own calendar and festivals based on local customs and beliefs. In that spirit, while it makes sense to try and emulate some of the festivals from the calendars we know most about, it also makes the most amount of sense to take local and personal customs and import those into your Hellenic practice.
For that reason, some of us honor some of these holidays while others do not for they have no particular meaning. My own calendars surround both the solar and lunar cycles: new moons for rites to Hekate (which was the practice in ancient times), full moons for Diana and/or Artemis, equinoxes for the ascent and descent of Persephone into the underworld and honoring her mother Demeter accordingly–which mirrors not only the mythos surrounding the two but also the Eleusinian Mysteries, and solstices for Apollo and Dionysos (summer and winter respectively) in a similar fashion as they exchanged at Delphi.
As far as potentially incorporating the Neo-Pagan Wheel of the Year, I also celebrate Hestia at Imbolc, Aphrodite at Beltaine, Apollo and Demeter at Lammas, and Persephone in her role as having assumed her throne in the underworld at Samhain. This is me personally according to my own customs and what makes the most amount of sense to me and as always, YMMV.
I would also suggest taking secular holidays of meaning and incorporating them. Holidays like Veteran’s Day would be great for Athena and Ares for instance, while Memorial Day would be a great day to honor Athena and Hermes in his role as guide of the dead. If you have local or family holidays or customs, those are also fairly easy to figure out ways to honor the Greek gods during those times.
It is both interesting and not entirely unexpected that in order to be true to the spirit of the ancients, we must make up our own calendars. Some holidays and traditions may be similar across the board, and some will not. You may even find a local group or start one which agrees upon and follows an established calendar. Either way, I think while it’s useful to know about Athenian holidays, taking them verbatim and trying to shoehorn them into our own lives actually doesn’t make sense, and wouldn’t be true to the spirit of how calendars and festivals were done back then. To use them as a template or a means of getting good ideas, certainly–and if certain holidays such as the Anthesteria call to you, by all means make them a part of your personal calendar!
I have some resources on ancient Greek holidays on my recommended reading list page.