Thanks to both Dom Stubbs & Media Girl. You spurred me to deeper investigation and I've got a solution working for me.
It looks like when the 2.5.3 update started encoding the Live Look links in my control panel, it exposed a problem with my server not handling those links properly. What I failed to mention was the error the link was showing in my browser which was "The URI you submitted has disallowed characters."
Before the Live Look URL was encoded, I didn't have this issue, but now I do, or did.
I've had the particular site I'm speaking of on Dreamhost which seemed to only work properly when I'd set the "uri_protocol" in my config file to "QUERY_STRING". My setting of the uri_protocol was related to my desire to get rid of the "index.php" in my URLs. I worked on the htaccess file so long ago, I don't recall the exact reasoning behind where I ended up.
Regardless of why I got to the point of use "QUERY_STRING", I set it back to "AUTO", which is the default, and my LIVE LOOK links are working in the new encoded version. I suspect something has changed with my Dreamhost PHP set up in some automatic update that I didn't notice that allowed me to stop using "QUERY_STRING."
So, in short, changing "uri_protocol" in my config file from "QUERY_STRING" to "AUTO" solved the problem with the encoded URL not working for me.