This may not get this in all places in your theme, but I think it will after a quick glance at your theme files.
Open the file called
content.php in your theme folder.
Find the line with this:
<?php printf( __( 'by', 'ilisa' )); ?> <?php the_author(); ?> and delete that.
Next to it, you'll see a line with this:
<?php printf( __( 'on', 'ilisa' )); ?>
Change it to:
<?php printf( __( 'Posted on', 'ilisa' )); ?>
That should take away your byline, and make it say "Posted on [date here].
It's a good idea to make sure your theme files are backed up before editing. Also, edit them using a text editor, not in the Theme Editor within WordPress. That way, if you have issues, you can always go back to the previous version. :)
Let me know if you have questions!