Fix
Download Process Explorer here.
Start your application and reproduce runtime error R6034.
Start Process Explorer. In the "View" menu go to "Lower Pane View" and choose "DLLs".
In the top pane, locate your application and click on it. The bottom pane should show a list of DLLS loaded for your application.
Locate "msvcr??.dll" in the list. There should be several. Look for the one that is not in the "winsxs" folder, and make a note of it.
Now, check the path just before your application runs. If it includes the folder you noted in step 5, you've probably found the culprit.
How to fix the problem? You'll have to remove the offending entry from the path before running your program. In my case, I don't need anything else in the path, so I wrote a simple batch file that looks like this:
path=
myprogram.exe
That's it. The batch file simply clears the path before my program runs, so that the conflicting runtime DLL is not found.