Note: In the commands below you should replace [user] with your own username.
Here are the steps I took.
1. If you don't already have it, install Wine.
sudo yum install wine
5. Download Slingplayer from http://www.slingmedia.com and save it to your desktop. Thanks to Jeff (see the comments below) for pointing out that there are as yet unresolved issues with installing version 2.0 so for the time being I suggest doing what I did and choose 'Ireland' as your country on the Slingmedia download site - the default version for download here is still 1.5.
6. Download the dependencies Slingplayer requires to run under Wine from here and here and extract the files to a folder on your desktop.
7. Using your Nautilus file manager open the folder you created in the previous step and copy all the Dll files and paste them in your /home/user/.wine/drive_c/windows/system32 folder choosing to overwrite system files.
8. While still in the system32 folder choose the Terminal option from the File menu in your file manager and issue the following command to register the dll files you just added.
wine regsvr32.exe msxml3.dll qcap.dll quartz.dll
9. Open the Wine configuration window again as you did in step 2 and click on the Libraries tab.and add the following overrides. gdiplus.dll, msxml3.dl quartz.dll. Select gdiplus.dll, click edit and set it to native only. Do the same for msxml3.dll. Click 'Apply' at the bottom of the winecfg window.
10. Click on the Audio tab at the top of the Wine Configuration winodow and select to use the ALSA driver only. Set the Hardware Acceleration option to Emulation, the Default Sample Rate to 22050 and the Default Bits per Sample to 8. Check the Driver Emulation checkbox. Click apply
11. Click the Graphics tab and check the following boxes only. Allow the window manager to decorate the window, Allow the windows manager to control the windows and Enable a virtual desktop. Enter 1024 and 760 in the Desktop size boxes. In the Vertex shader support options choose Hardware and check the box for Allow pixel shader (if supported by hardware).. Click OK to exit winecfg.
12. Now we can install Slingplayer. Go back to your terminal window and change your working directory to your desktop with
wine /home/[user]/.wine/drive_c/Program\ Files/Sling\ Media/SlingPlayer/SlingPlayer.exe
14. You may now click the Play button and start streaming. I got a notice informing me that I needed to update the firmware in my Slingbox (which was a bit odd because it ran fine under my Windows installation of Slingplayer moments earlier) and finding that I could not continue streaming without doing so, I took a chance and did without issue.
15. One final optional step enables you to add a custom shortcut to launch Slingplayer to your main menu and consequently to the launcher.