Bash, 85 81 78 74 72 bytes

for((;i++<100;)){
o=($i Fizz Buzz FizzBuzz)
echo ${o[!(i%3)+2*!(i%5)]}
}

Thanks to @Neil for saving 4 bytes!
Thanks to @manatwork for saving 3 bytes!
Thanks to @primo for saving 2 bytes!

Try it online!

Bash, 85 81 78 74 72 71 bytes

for((;i++<100;j=i%3&2|i%5/4)){
o=($i Buzz Fizz FizzBuzz)
echo ${o[j]}
}

Thanks to @Neil for saving 4 bytes!
Thanks to @manatwork for saving 3 bytes!
Thanks to @primo for saving 2 bytes!

Try it online!

Bash, 85 81 78 74 bytes

for i in {1..100}
{
o=($i Fizz Buzz FizzBuzz)
echo ${o[!(i%3)+2*!(i%5)]}
}

Thanks to @Neil for saving 4 bytes!

 Thanks to @manatwork for saving 3 bytes!

Try it online!

Bash, 85 81 78 74 72 bytes

for((;i++<100;)){
o=($i Fizz Buzz FizzBuzz)
echo ${o[!(i%3)+2*!(i%5)]}
}

Thanks to @Neil for saving 4 bytes! 
Thanks to @manatwork for saving 3 bytes!
Thanks to @primo for saving 2 bytes!

Try it online!

Bash, 85 81 78 bytes

for i in {1..100}
{
((a=i%3,b=i%5))||i=
((a))||i=Fizz
((b))||i+=Buzz
echo $i
}

Thanks to @Neil for saving 4 bytes!

Thanks to @manatwork for saving 3 bytes!

Try it online!

Bash, 85 81 78 74 bytes

for i in {1..100}
{
o=($i Fizz Buzz FizzBuzz)
echo ${o[!(i%3)+2*!(i%5)]}
}

Thanks to @Neil for saving 4 bytes!

Thanks to @manatwork for saving 3 bytes!

Try it online!