"Dictatorships have no place in the Americas. May God bless the Cuban people who are struggling for freedom," Mr. Bush said.
The taped message to those living under the regime of Fidel Castro was aired Tuesday morning by Radio Marti, a U.S. government station beamed into Cuba.
The president also planned to meet later in the day with Cuban dissidents and former political prisoners to "hear their stories," said White House spokesman Ari Fleischer.
Among those Mr. Bush will receive is Ramon Colas, who started an independent library movement in Cuba and left the island after undergoing severe harassment by the authorities.
In his 40-second radio message, Mr. Bush said: "On behalf of the people of the United States, I send greetings to the Cuban community. My hope is for the Cuban people to soon enjoy the same freedoms and rights as we do."
Tuesday was the 101st anniversary of Cuban independence from Spain. The Cuban government does not regard May 20 as Independence Day, contending that Cuba did not achieve true self-determination until the 1959 revolution.
With the anniversary in mind, Cuban-American and other groups have been lobbying the White House for changes in policy toward Cuba.
The most powerful of the anti-communist exile groups, the Cuban-American National Foundation, urged the administration to adopt a regime change policy for Cuba.
It also called for "massive" assistance to democracy advocates in Cuba and for the indictments of Castro ands his brother, Raul, for their alleged role in the shooting down of two private, unarmed Miami-based planes north of Cuba in 1996.
Other Cuban-American groups favored stepping up economic pressure on the regime by sharply restricting the flow of dollars to the island from U.S. relatives and friends.
From the liberal side, the Washington Office on Latin America recommended lifting restrictions on travel to Cuba by Americans and to pursue a policy of engagement with the island. They see these steps as the best hope for improving the country's human rights conditions.