I was having this problem too.
Make sure your permissions for the designated folder are set to 755.
I went into Settings/Misc and unchecked the box for “Organize my uploads into month- and year-based folders”.
But at the same time I added a .htaccess folder to wp-admin with the code below. Now I can upload images but the do not appear on the initial post page. I’m not sure which action worked but it did. So try one and then the other and test after each idea.
<Limit GET POST>
order deny,allow
deny from all
allow from all
</Limit>
<Limit PUT DELETE>
order deny,allow
deny from all
</Limit>
# BEGIN WordPress
<IfModule mod_security.c>
SecFilterEngine Off
SecFilterScanPOST Off
</IfModule>
# END WordPress
AuthName drmomonline
IndexIgnore .htaccess */.??* *~ *# */HEADER* */README* */_vti*