What three amendments to the US Constitution dealt with suffrage during the 1960s and 1970s?

During the time that you mention there were three
amendments that had to do with the right to vote.

First,
there was the 23rd Amendment, passed in 1961.  This amendment gave the residents of
Washington, D.C. the right to vote in presidential elections.  Second, there was the
24th Amendment.  This barred states from imposing a poll tax as a requirement for being
allowed to vote.  This practice had been used to prevent blacks from voting in the
South.  Finally, there was the 26th Amendment, passed in 1971.  This amendment expanded
the franchise to give people who were 18 years or older (but not yet 21) the right to
vote.

Between them, these three amendments helped to make
America more democratic.