Ver. 2

 
#!/bin/bash --posix
IFS=''
while read -r -N 1 char
     do
       case $char in
         [[:lower:]])
                  case $char in
                        "а")    printf "%s" "a";;
                        "о")    printf "%s" "o";;
                        "е")    printf "%s" "e";;
                        "р")    printf "%s" "p";;
                        "с")    printf "%s" "c";;
                        "у")    printf "%s" "y";;
                        "х")    printf "%s" "x";;
                        *)      printf "%s" $char;;
                  esac
       ;;
       [[:upper:]])
                   case $char in
                        "А")   printf "%s" "A";;
                        "Е")   printf "%s" "E";;
                        "О")   printf "%s" "O";;
                        "В")   printf "%s" "B";;
                        "К")   printf "%s" "K";;
                        "М")   printf "%s" "M";;
                        "Н")   printf "%s" "H";;
                        "Р")   printf "%s" "P";;
                        "С")   printf "%s" "C";;
                        "Т")   printf "%s" "T";;
                        "Х")   printf "%s" "X";;
                        *)     printf "%s" $char;;
                     esac
        ;;
        *) printf "%s" "$char";;
   esac
done

Due to conflicting historical implementations of the echo command, POSIX®
recommends that printf is preferred over echo.