How can I run just the statement my cursor is on in SQL Server Management Studio?

Use Ctrl+KU to select a line. Then use F5 to run it.

Although it only works for single line selection, still I find it quite useful.

Hope it helps!!

You can check out this add-in for SSMS 2012. Place the cursor within the statement you want to execute and press CTRL+SHIFT+E

SSMS Executor - https://github.com/devvcat/ssms-executor/releases

Update:
Project moved to github and the addin re-written to support SSMS 2014, SSMS 2016. (Previously, the project lived on codeplex, at SSMS Executor - http://ssmsexecutor.codeplex.com/.)

Ok, so what I'm getting from all these answers is "No, that is not possible."

Edit:

Here is how I was able to do this:

1 - Download SQL Developer

2 - Download the jTDS driver

3 - Follow these instructions to add that driver to SQL Developer

4 - Connect to SQL Server using SQL Developer (cool!)

5 - Run it and life is good