Building on the answer by @JoelCoehoorn, my approach is to leave all my PRINT statements in place, and simply follow them with the RAISERROR statement to cause the flush.
PRINT 'MyVariableName: ' + @MyVariableName
RAISERROR(N'', 0, 1) WITH NOWAIT
The advantage of this approach is that the PRINT statements can concatenate strings, whereas the RAISERROR cannot. (So either way you have the same number of lines of code, as you'd have to declare and set a variable to use in RAISERROR).
If, like me, you use AutoHotKey or SSMSBoost or an equivalent tool, you can easily set up a shortcut such as "]flush" to enter the RAISERROR line for you. This saves you time if it is the same line of code every time, i.e. does not need to be customised to hold specific text or a variable.