Don't remember where I got this from but credit goes to where it's deserved, this dumps the databases into a directory structure like so: /MY_Database_Backup/YYYYMMDD/database1.gz
It will also automatically delete the old ones.

#!/bin/bash
#

# Modify the variables below to your need

# Mysql Credentials
MyUSER="mysql_username"
MyPASS="mysql_password"
MyHOST="myhost"

# Owner of mysql backup dir
OWNER="owner_of_backup_dir"
# Group of mysql backup dir
GROUP="group_of_backup_dir"

# Which databases to backup
DBS="list specific databases with spaces"
# Or get all databases
#DBS="$($MYSQL -u $MyUSER -h $MyHOST -p$MyPASS -Bse 'show databases')"

# DO NOT BACKUP these databases
IGGY=""

# Backup Dest directory, change this if you have someother location
DEST="/my/backup/dir/"

# mysqldump parameters
# YOU MAY HAVE TO CHANGE THIS PARAMATER
# This worked for me
DUMP_OPTS="--max_allowed_packet=1G -Q --single-transaction"

# Send Result EMail
# SEND_EMAIL = 1=yes 0=no
SEND_EMAIL=1
NOTIFY_EMAIL="my_email@domain.com"
NOTIFY_SUBJECT="MySQL Backup Notification"

# Delete old backups
# DELETE_OLD_BACKUPS= 1=yes 0=no
DELETE_OLD_BACKUPS=1
DELETE_BACKUPS_OLDER_THAN_DAYS=7

# Usually there is no need to modify the variables below

# Linux bin paths, change this if it can't be autodetected via which command
MYSQL="$(which mysql)"
MYSQLDUMP="$(which mysqldump)"
GREP="$(which grep)"
CHOWN="$(which chown)"
CHMOD="$(which chmod)"
GZIP="$(which gzip)"
MAIL="$(which mail)"
FIND="$(which find)"
DF="$(which df)"

# Get hostname
HOST="$(hostname)"

# Get data in yyyy-mm-dd format
NOW="$(date +"%Y%m%d")"

# Function for generating Email
function gen_email {
DO_SEND=$1
TMP_FILE=$2
NEW_LINE=$3
LINE=$4
if [ $DO_SEND -eq 1 ]; then
if [ $NEW_LINE -eq 1 ]; then
echo "$LINE" >> $TMP_FILE
else
echo -n "$LINE" >> $TMP_FILE
fi
fi
}

# Main directory where backup will be stored
if [ ! -d $DEST ]; then
mkdir -p $DEST
# Only $OWNER.$GROUP can access it!
$CHOWN $OWNER:$GROUP -R $DEST
$CHMOD 0750 $DEST
fi

# Create backup directory
MBD="$DEST/$NOW"
if [ ! -d "$MBD" ]; then
mkdir "$MBD"
# Only $OWNER.$GROUP can access it!
$CHOWN $OWNER:$GROUP -R $MBD
$CHMOD 0750 $MBD
fi

# Temp Message file
TMP_MSG_FILE="/tmp/$RANDOM.msg"
if [ $SEND_EMAIL -eq 1 -a -f "$TMP_MSG_FILE" ]; then
rm -f "$TMP_MSG_FILE"
fi

set -o pipefail

# Start backing up databases
for db in $DBS
do
skipdb=-1
if [ "$IGGY" != "" ];
then
for i in $IGGY
do
[ "$db" == "$i" ] && skipdb=1 || :
done
fi

if [ "$skipdb" == "-1" ] ; then
FILE="$MBD/$db.$HOST.$NOW"
# do all inone job in pipe,
# connect to mysql using mysqldump for select mysql database
# and pipe it out to gz file in backup dir
$MYSQLDUMP $DUMP_OPTS -u $MyUSER -h $MyHOST -p$MyPASS $db | $GZIP -9 > "$FILE.gz"
ERR=$?
if [ $ERR != 0 ]; then
NOTIFY_MESSAGE="Error: $ERR, while backing up database: $db"
else
NOTIFY_MESSAGE="Successfully backed up database: $db"
fi
gen_email $SEND_EMAIL $TMP_MSG_FILE 1 "$NOTIFY_MESSAGE"
echo $NOTIFY_MESSAGE
fi
done

# Empty line in email and stdout
gen_email $SEND_EMAIL $TMP_MSG_FILE 1 ""
echo ""

# Delete old backups
if [ $DELETE_OLD_BACKUPS -eq 1 ]; then
find "$DEST" -maxdepth 1 -mtime +$DELETE_BACKUPS_OLDER_THAN_DAYS -type d | $GREP -v "^$DEST$" | while read DIR; do
gen_email $SEND_EMAIL $TMP_MSG_FILE 0 "Deleting: $DIR: "
echo -n "Deleting: $DIR: "
rm -rf "$DIR"
ERR=$?
if [ $ERR != 0 ]; then
NOTIFY_MESSAGE="ERROR"
else
NOTIFY_MESSAGE="OK"
fi
gen_email $SEND_EMAIL $TMP_MSG_FILE 1 "$NOTIFY_MESSAGE"
echo "$NOTIFY_MESSAGE"
done
fi

# Empty line in email and stdout
gen_email $SEND_EMAIL $TMP_MSG_FILE 1 ""
echo ""

# Add disk space stats of backup filesystem
if [ $SEND_EMAIL -eq 1 ]; then
$DF -h "$DEST" >> "$TMP_MSG_FILE"
fi
$DF -h "$DEST"

# Sending notification email
if [ $SEND_EMAIL -eq 1 ]; then
$MAIL -s "$NOTIFY_SUBJECT" "$NOTIFY_EMAIL" "$TMP_MSG_FILE"
rm -f "$TMP_MSG_FILE"
fi