Tim Thumb image problems with MultiSite

I'm moving a single WordPress site into a MultiSite network and having a problem with the images. This is an archive on the single site (still up and running):


And this is the same page on the network site:


The error presented when I right click on the images on the network site and choose "View image" is:

A TimThumb error has occured
The following error(s) occured:

    Could not create the file cache directory.

Oddly enough, the main site on the network displays images just fine, and both sites are using child themes of the same theme framework. Here's the main site on the network:


I've tried to figure out where the cache directory is supposed to be for the problem site and create it myself, but still no joy. Since the problem site is using a child theme just like the main site is, you'd think that the cache directory would be the same one, and since the main site is working, that the problem site would be working too, but it's not obviously.

In the past, I've had to ask my host to set special ownership and permissions for cache and upload directories for them to be functional for Wordpress, and this may be necessary for this problem as well, but I'd like to make sure that I've not set something up incorrectly before I trouble them with this, and also, I don't know which directory to tell them to set permissions and ownership on at this point.

If a WP expert could have a look and help me out, I'd appreciate it!