But, since message is now a Row[...], the concatenation can't go through since it's no longer a string

You can try Join, like this, to see if it works for you

nextMessage = Style["the oven is getting hot!", Red];
message = Row[{"Would like to say that ", importantMsg, " and needs cooling"}];
MessageDialog[message];

and now

nextMessage = {"..please hurry"};
message = Row[Join[First@message, nextMessage]];
MessageDialog[message];

and now

nextMessage = {Style["..I am still waiting..", Red]};
message = Row[Join[First@message, nextMessage]];
MessageDialog[message];

Nasser, thanks for the quick reply!

That's already a help, but with still this lingering issue...as the main program keeps running, it keeps track of a cumulative message string (continually updated): i.e., if the current message (to be printed via MessageDialog) is your message = Row[{"Would like to say that ", importantMsg, " and needs cooling"}];, I still need - at some later point - to have something like message <> "here is another string (possibly with its own highlighting)" to be available to MessageDialog. But, since message is now a Row[...], the concatenation can't go through since it's no longer a string.

Hope that makes sense, and my problem is clear. Essentially I need what your great suggestion does for this one message, but I need to be able to keep concatenating strings to this (some of which may have highlights, some not).