(CNN) -- Tiempo Libre named their new CD "What You've Been Waiting For/Lo Que Esperabas." And this week, the guesswork is easy: The seven members of Tiempo Libre are waiting for their Grammy.

In contention on Sunday for a best tropical latin album trophy, the Miami, Florida-based Cuban collective knows the ropes, both in spirit and in business terms.

They're on their second Grammy trip, having been nominated last year for their CD, "Arroz con Mango," a slang phrase for something amazing.

And win or no win this year, they head on to concert dates in Hawaii, then back to Florida to close the Festival of the Arts Boca in Boca Raton on March 11 and a continually lengthening slate of domestic and international bookings, "including Hong Kong, Singapore, Malaysia --" music director Jorge Gomez breaks off in his list: "In Asia, they love us, they now are speaking Spanish there like crazy people!"

That laugh, a yelp of happiness that could come right out of a Tiempo Libre recording session, is as good a clue as any to their signature sound: timba, pronounced "TEEM-bah."

Rooted in the classics and driven by Havanan tradition, these musicians are firming up a distinctive voice of the Cuban diaspora, and it's inevitably exuberant, surging with determination, the sound a fast-rising tide of adamant pride.

"Timba, you know," Gomez says, searching for his words, "it's a mix. It's jazz and son." That's the strain of salsa born in Cuba's Oriente province and first heard in Havana in the late 19th century.

"When you get that mix, it sounds like 'tah-bababala-yow' and then 'toon-get-tacka-teek.' "

Separate journeys

"When you talk about music here in Miami," Gomez says, "it's a small city. Everybody knows everybody. 'Hey, you know who's here? Yeah, Joachin is here now!' And that's how we got together."

The son of two Cuban classical pianists, Gomez, 36, stops laughing only when he talks about how the group's members reached that "small city" in the United States.

"The funny thing about it," he says, "is that we were all in school together. We knew each other in Havana." They were at the ENA, the Escuela Nacional de Arte, National School of Art, in classical music. Some, like trumpeter Pavel Diaz, studied in Havana's Alejandero G. Caturla elementary school and had gone on to work in the Amadeo Roldan conservatory there.

But when it came time for each of them to leave Cuba, the friends used different routes. Bassist Tebelio (Tony) Fonte Pedraja, for example, went to Berlin and played there with Todos Estrellas, Seformo and Latin Spektrum, before arriving in Miami in 2002.

Sax and flute man Luis Beltran Castillo started his music studies at age 11 in his hometown, Pinar del Rio, before moving to Havana and the ENA -- then toured with groups in Europe before getting to the United States. Drummer Hilario Bell and vocalist Joaquin Diaz, both from Santiago de Cuba, also had roundabout journeys to the United States.

"Nobody leaves Cuba and goes straight to the United States," Gomez says. "All of us spent five, 10 years in other countries -- Italy, Germany, Mexico. It was not so difficult for me. I was in the army in Cuba. I said, 'Hey, I have to go see my family in Guatemala.' They said, 'Sure, go, go.' And I never came back."

Once they'd found each other in Miami, they began playing together as an off-hours sideline in their free time -- tiempo libre. When their first CD, Manos Pa'rriba ("Hands in the Air") drew them a multi-CD contract with the Shanachie label, they knew they had something.

Today, a hope of playing on the island again is on the minds of these musicians, Gomez says, amid reports of Fidel Castro's uncertain health. (See video of the Cuban president released at the end of January.) "Soon, maybe," Gomez says.

Meanwhile, the group's reception and rising Grammy presence buoy them like the ebullience of the music they produce in two stage shows, one for primarily English-speaking audiences, and one for Spanish-speaking fans.

"We like to teach," says Gomez of himself and his former-student buddies, "and not just music but also about dancing it -- salsa, conga, rumba."

A collaborative work called Rumba Sinfonica is in the offing with Venezuelan-born composer Ricardo Lorenz, of the Chicago Symphony's residency program. The work is being created for symphony and Tiempo Libre for a premiere, Gomez says, in Detroit in November.

And that classical connection, still so strong for these boyhood friends, echoes the group's debut in 2002 north of Chicago at the revered Ravinia Festival, which currently touts spring bookings with pianist Adam Golka, the Enso String Quartet and violinist Christina Castelli. Gomez and his colleagues opened in 2002 for the great son-Cubano artist Celia Cruz, who would die a year later. (Read an account of Cruz's funeral by CNN's Rose Arce.)

"Everybody is with us," Gomez says, keenly aware of the subtle shades of Latin musical influence being kept alive and thriving in his band's work. The laugh is there, but softer now.

"When we go to the Grammys, you know, we go with everybody. We go with Cuba."