Finally found an answer to this on this thread.
There are two suggestions. The first didn't work for me, but the second one did.
It also looks like the code has changed a little since that answer, so I just copied the end of the new code into the replacement code, although the orignal replacement code also worked.
I'll go ahead and give the solution that worked for me.
Go to wp-admin/edit.php and find the lines
wp
("what_to_show=posts$post_status_q&posts_per_page=15&order=$order&orderby=$orderby");
You can delete that, or even better just comment it out so that it's still there if you need to change things back. You can comment it out by putting /* at the beginning and */ at the end so that it looks like this:
/*wp("what_to_show=posts$post_status_q&posts_per_page=15&order=$order&orderby=$orderby");*/
Now put in the new replacement code:
$thisauthor='';
if (!current_user_can('edit_others_posts'))
$thisauthor = '&author=' . $user_ID;
wp('what_to_show=posts&posts_per_page=15&order=$order&orderby=$orderby' . $thisauthor);