What is True or not is off-topic for the site, but I can tell you what the common teaching is from a Young Earth Creationist perspective. Note that the YEC view is not the only view Christians hold, but it is clearly the one you are looking for.
To potential commentators, remember that whether this is true or not is off-topic, and debate in comments is not constructive.
Most adherents to the literal young-earth seven-day creation account believe that when Adam and Eve ate of the fruit, God applied the curse to the entire universe, including animals. For example, from http://www.answersingenesis.org/articles/kw/museum-guide-corruption
God had warned Adam and Eve that if they ate of the Tree, they would
die. When God came to walk with them that evening, He punished their
sin. The earth was now cursed. Death was now part of life; both
animals and humans would now die (Genesis 3:19; Romans 8:20–22). When
the first humans sinned, it corrupted all of creation. The ground was
cursed and would produce thistles and thorns (Genesis 3:17–18).
Animals began to hunt and eat other animals. Man would now have to
work hard for food, and woman would have pain in bearing and raising
her children (Genesis 3:16–17).
The Institute for Creation Research (ICR) says this:
The animals, not having moral natures, were not guilty of sin, of
course, but they also shared in the Curse, for they were—like
Adam—made of the dust of the ground that God had cursed. Like Adam's
body, their bodies also must return to the ground. Henceforth, "death
reigned ... even over them that had not sinned after the similitude of
Adam's transgression" (Romans 5:14). Thus there is death in the world
only because there is sin in the world.