I like Moro's suggestion to hit it on a country by country basis, kinda like a greatest hits album or something. But i think you need a representation across the flavor and strength spectrums (plus a good sample of wrapper types) as most of the stuff listed previously are stronger than many Cubans. Here's my two cents:
Dominican Republic - Davidoff (Special Series or Millennium Blend), AVO Domaine, La Aurora Preferidos Cameroon, La Flor Dominicana (Cameroon Cabinet, Double Ligero, or LG Diez) - sorry not a big Fuente fan
Honduras - Hoyo de Tradicion from HdM, Camacho (the Diploma line has the best quality), maybe a Carlos Torano, Rocky Patel in there too (some are made in Honduras). And you might as well try a Puros Indios (a very unusual taste profile - many do not like it)
Nicaragua - ah, this is where the hottest cigars are coming from... Oliva, Padron (the Maduros taste better to me), La Riqueza and Cabaiguan. Go ahead and try Joya de Nicaragua (an Antano) while you're at it.
Bahamas - Graycliff (try the Red Label and the Espresso for the range in flavor)
Costa Rica - Bucanero - my recommendation would be the Treasures of Costa Rica cigar (it's a puro), or the Canon Cubano or Don Douglas
Brazil - Most likely will have to try a Dona Flor (puro) or many maduros use a Brazilian Mata Fina or Aripiraca wrapper (CAO Brazilia or Sopranos or even a Butera Royal Vintage - whichever is the Maduro one...)
Columbia - Bravo Cigars Columbian Gold (only puro I know)
Mexico - many use Mexican leaves as binder, but to get the Mexican experience I would suggest a Te-Amo or A. Turrent. Mexican tobacco is a love or hate thing. Most hate it.
Ecuador - haven't tried any cigars MADE in Ecuador (although there is one called Capa Fina), but there are many that feature wrappers from there - mostly Ecuadorian Connecticut and Sumatra varieties.
USA - we're now talking boutiques here... Tatuaje Cazadores, Don Pepin White Labels, El Titan de Bronze, Moore & Bode, La Luna and the list goes on. Also our two big leaves are Connecticut Shade and Connecticut Broadleaf - these can be found on TONS of cigars.
I would recommend you try Puros first, then some of the blends. Plus there are a ton of other wrappers you must try. Go to my blog here (
http://zencigar.wordpress.com/the-art-of-the-cigar/the-definitive-i-hope-cigar-wrapper-guide/) for some more examples to try and other wrapper types.
I would also say not everything listed above would hold a candle to a good Cuban, but there are many good cigars listed above, and I would agree with many in the lists the other Brothers have made for high QUALITY examples of Non-Cubans.
I mostly put this list out so that you can get an idea of the range in taste (and quality) from country to country if you are into that.