The drink is not necessarily chilled.
Many variations are appropriate too.
The one as described in the recipe.
Another, the same without ice. And another version more like a caipirinha, with more lime and with or without ice.

Some recipes allow nearly only peel, not flesh. Small piece, no ice, little syrup. Lot of rhum.

This is really intriguing. I've never seen the Maurin Quina before and will definitely look for it. I'm wondering what would happen if Cherry Heering or Combier Rouge were substituted for the cassis here.

I tend to think about cocktail development in terms of flavor or aroma accords. In this cocktail, I was thinking about the bitters first - they're fruity, with a chile spicyness and typical bitter notes. That led me to Cassis, which is fruity + a woody bitterness, which led me to Rye, which has a green, woody bitterness as well. Maurin Quina is cherry + almond + some interesting bitter notes, and the rinse of Creme de Cacao is meant to mimic the 20th Century cocktail, as well as to tie in and harmonize the other flavors, all of which go well with chocolate. This is still in the theoretical phase, as Maurin Quina is a special order only thing where I live ;) 

But to answer your question... I think that other cherry liqueurs might be a good substitition, as long as they have a bitterness to them. You might also want more lemon juice in the cocktail to make up for the loss of acidity in the cherry liqueurs.

So I'm busy today. My likker store actually sourced a bottle of Maurin Quina for me, and I made the drink as written (the bitters are dropper instead of dash). I like this, if I do say so myself. It tastes like a good left bank Bordeaux in a weird way, with woody notes forward, cassis and red fruit in the midpalate, then the bitterness of the Maurin Quina mixed with the chile-accented bitters on the finish. I think a bit of extra shaking is a good thing as it brings out the brambly Cassis more.

Quite tasty, but I'd cut the bitters to one dash, otherwise it overwhelms the other ingredients.

I have sought confirmation for Elixir about the name, ingredients, history, and attribution for this cocktail. This is obviously not a relative of a true Martini and perhaps has a better name.

Sounds similar to the Dreamy Dorini Smoking Martini created by Audrey Saunders that I saw mentioned on eGullet:
http://forums.egullet.org/index.php?/topic/82754-simple-vodka-cocktail-…

H. Joseph Ehrmann kindly wrote back with the original recipe, notes, and an aritcle from November 30, 2008 in the New York Times where it was featured. I have updated the recipe's name, history, date, and reference to reflect these source references.

Thanks, Dan, for doing that, and thanks to H. Joseph Ehrmann.

This is a perfectly balanced drink. I've been making it with Eagle Rare 10, though will be trying with Weller 107 -- I'm curious to see how it performs with a higher-proof, wheated spirit.

I've made this now with both bourbon and rye. It's an excellent drink in both iterations: spicy, dark, strong, not too sweet (though maybe still too sweet for some), integrated and balanced. The rye version wins by a hair for me as it makes a dryer drink and its spice plays well with the bitters. (not having celery bitters, I went in a different direction and used Dutch's Colonial, which have a woodsy and anise-y character that plays well with Cynar and rye. Not sure whether that changes the cocktail enough to merit a new name.) The rye and Cynar open the sip while the Maraschino lingers in the finish. The rye draws some cool dark cherry notes out of the Cynar and the two together develop an almost cola taste, but an herbal, 19th Century cola. Moderately but pleasurably bitter. Recommended for fans of the New York-themed Bitter/Brown/Stirred standbys (Little Italy, Red Hook, etc). My only complaint is its name: a great drink deserves a better one. 4.5/5

A little too sweet. 

See also the Pink Lady, which includes egg white.

Curated to add credit for this adaptation, per reference link. Thanks @dustinw84

I keep coming back to this one as we transition into fall. So good! I cut the grenadine to 1/2 and I think it's improved, for my taste.

Solid drink. I would cut back the sweet vermouth to an ounce, though.