There is truth to it! However, the proper way to think of this is on a state-by-state level. As an example, though, New York State's second constitution, adopted in 1821, dropped the property requirement for white men, but retained it for black men, at Article II, Section 1:

no man of colour, unless he shall have been for three years a citizen of this state, and for one year next preceding any election, shall be seized and possessed of a freehold estate of the value of two hundred and fifty dollars, over and above all debts and incumbrances charged thereon; and shall have been actually rated, and paid a tax thereon, shall be entitled to vote at any such election.

Moreover, I would argue that the "property" requirement didn't disappear entirely until 1965, when Section 10 of the Voting Rights Act (pdf) stated Congress's finding that poll taxes -- requirements that voters pay a nominal sum before being allowed to vote -- were unconstitutional, and instructed the Attorney General to sue every state that continued to maintain them. Before that date, poll taxes ensured that poor (and disproportionately black) citizens would not be able to vote, because they could not afford the tax. This is not a property tax in the conventional sense that it excludes all but landholders from the franchise. But it is a property tax in the sense that it kept people who couldn't afford a certain standard of living from participating in democracy. Robert Caro's Master of the Senate has a good (and lengthy) prologue about Jim Crow, and other impediments to the exercise of voting rights.

The short answer, then, is that voting and voting rights were entirely different creatures in early America, when compared to today. If you have another more specific inquiry I would be happy to expand.