In bash version 3 you can use the '=~' operator:
if [[ "$date" =~ ^[0-9]{8}$ ]]; then
echo "Valid date"
else
echo "Invalid date"
fi
Reference: http://tldp.org/LDP/abs/html/bashver3.html#REGEXMATCHREF
NOTE: The quoting in the matching operator within the double brackets, [[ ]], is no longer necessary as of Bash version 3.2