My short yet readable Python version:

    for i in range(1,101):
        if not i % 15: print 'FizzBuzz'
        elif not i % 5: print 'Buzz'
        elif not i % 3: print 'Fizz'
        else: print i

The same in Bash:

    for i in {1..100}
    do
        let "$i % 15" || { echo "FizzBuzz"; continue; }
        let "$i % 5" || { echo "Buzz"; continue; }
        let "$i % 3" || { echo "Fizz"; continue; }
        echo $i
    done