Your bash example could be a little cleaner. Since 0 is truthy in bash and [ `expr something` ] is, for most purposes, (( )).

    for i in {1..100}; do
        if (( $i % 3 && $i % 5 )); then
            echo FizzBuzz
        elif (( $i % 3 )); then
            echo Fizz
        elif (( $i % 5 )); then
            echo Buzz
        else
            echo $i
        fi
    done

Bash also has a decently powerful matching statement so you can do something similar to the rust example.

    for i in {1..100}; do
        case "$(( $i % 3 ))$(( $i % 5 ))" in
            00)
                echo FizzBuzz
                ;;
            0*)
               echo Fizz
               ;;
            *0)
               echo Buzz
               ;;
            *)
               echo $i
               ;;
        esac
    done