(1) Certain people felt a man of Lincoln's importance deserved to be on a higher denomination than a penny. The counter argument to this was that Lincoln was the common people's president; therefore the penny was the appropriate choice.
(2) Some citizens, including many Southerners, were disturbed that putting the features of a president on a coin was too reminiscent of European monarchies. It didn't "fit" with their idea of U.S. democracy.
(3) Criticism was directed at the replacement of the Indian Head cents. These had been in circulation since 1859 and were one of the most famous of all United States coins.
(4) There was a major outcry over the placement of Victor David Brenner's initials (V.D.B.) on the backside of the coin. Brenner was the Lincoln penny's designer. The protest was so loud that President William H. Taft's Secretary of the Treasury ordered all initials except the "B" removed within one week of the original issue. Still, over 28,000,000 of the "V.D.B." pennies were struck. Later, Brenner's initials were moved from the rear of the coin, where they had been clearly visible, and which was what started the protest, and placed below the "bevel" of the bust, slightly to the left of center when looking at the coin, in letters so small you need a very strong magnifier and an almost new penny to see them. But they are there, on every penny minted from 1919 to the present day.