Custom “radio button meta box” not saving correctly

The error is in you saving function where you are looking for $_POST['meta_box_musicreleases_linktype'] which is never set and should actually be $_POST['linktype'], and the use of wp_kses is not right here, if you want to validate the data and you know all of the accepted values the you can simply use in_array try:

add_action( 'save_post', 'musicreleases_linktype_meta_box_save' );
function musicreleases_linktype_meta_box_save( $post_id ){
    if( defined( 'DOING_AUTOSAVE' ) && DOING_AUTOSAVE ) return;
    if( !isset( $_POST['meta_box_nonce'] ) || !wp_verify_nonce( $_POST['meta_box_nonce'], 'my_meta_box_nonce' ) ) return;
    if( !current_user_can( 'edit_post' ) ) return;

    //accepted values whitelist
    $allowed = array('free','itunes','none');

    if( isset( $_POST['linktype'] )  && in_array($_POST['linktype'], $allowed))
        update_post_meta( $post_id, 'meta_box_musicreleases_linktype',  $_POST['linktype'] );
}

this way the data will be saved only if the value is in the accepted “whitelist” and anything else will be avoided.

Update

to show the selected value on the backend you need to fix you metabox function which currently i have no idea what you tried to do there but here is a simple fix

function musicreleases_linktype_meta_box_cb( $post )
{
    $value = get_post_meta( $post->ID,'meta_box_musicreleases_linktype',true );
    wp_nonce_field( 'my_meta_box_nonce', 'meta_box_nonce' );
    ?>
    <p>
        <input type="radio" name="linktype" value="freedownload_album" <?php echo ($value == 'freedownload_album')? 'checked="checked"':''; ?>/> Free<br />
        <input type="radio" name="linktype" value="buyonitunes_album" <?php echo ($value == 'buyonitunes_album')? 'checked="checked"':''; ?>/> iTunes<br />
        <input type="radio" name="linktype" value="none_album" <?php echo ($value == 'none_album')? 'checked="checked"':''; ?>/> None<br />
    </p>
    <?php   
}