How do I flush the PRINT buffer in TSQL?

Use the RAISERROR function:

RAISERROR( 'This message will show up right away...',0,1) WITH NOWAIT

You shouldn't completely replace all your prints with raiserror. If you have a loop or large cursor somewhere just do it once or twice per iteration or even just every several iterations.

Also: I first learned about RAISERROR at this link, which I now consider the definitive source on SQL Server Error handling and definitely worth a read:
http://www.sommarskog.se/error-handling-I.html

Another better option is to not depend on PRINT or RAISERROR and just load your "print" statements into a ##Temp table in TempDB or a permanent table in your database which will give you visibility to the data immediately via a SELECT statement from another window. This works the best for me. Using a permanent table then also serves as a log to what happened in the past. The print statements are handy for errors, but using the log table you can also determine the exact point of failure based on the last logged value for that particular execution (assuming you track the overall execution start time in your log table.)

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.

For example:

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.

Yes... The first parameter of the RAISERROR function needs an NVARCHAR variable. So try the following;

-- Replace PRINT function
DECLARE @strMsg NVARCHAR(100)
SELECT @strMsg = 'Here''s your message...'
RAISERROR (@strMsg, 0, 1) WITH NOWAIT

OR

RAISERROR (n'Here''s your message...', 0, 1) WITH NOWAIT