I'm also using a theme that wasn't widget-friendly but I changed it myself. Now I have a sidebar and a 'widget' tab in my admin panel.
The trick is to create a 'dynamic sidebar' by adding a functions.php file to your new theme directory using whatever file transfer software you have.
Step 1: (skip if your theme already has a function.php file) To do this, I found another theme that came with the functions.php file, copied it, and uploaded it to MY theme's directory.
Step 2: Replace sidebar.php coding (from the wp.org admin panel) to (copy and paste exactly as-is):
<?php if ( !function_exists('dynamic_sidebar')
|| !dynamic_sidebar() ) : ?>
<p>This is my blog.</p>
<?php endif; ?>
This makes your side a 'dynamic sidebar' so you can now use widgets.