I do it fairly often on my older bodies that don't have the anti-dust features. But before you try that, did you try to manually run the sensor cleaner that the camera has? I've found that to work very well and avoids the need to wet clean it.
take initial reference shot (white wall, f22)
hold camera with sensor facing down
manually evoke the sensor cleaning function
repeat 4-5 time
take new picture and compare to reference shot to see if it came clean
If that doesn't work then I will wet clean my sensor which is much easier than most claim but I will caution that you DO NOT WANT TO PUT MUCH PRESSURE ON THE SENSOR STACK. The new sensors have very thin filters on them that are sealed to the sensor so if you push too hard you will crack the filter. The good news is you hardly need to put any pressure on it as you are just picking up dust and tiny particles.
So, how to clean the sensor wet...
Get these....lens tissues
get some of these...
put the eraser on the end of a pencil
tightly wrap the tissue around the eraser
apply one drop of lens cleaning solution to the tip of the tissue
gently clean the entires surface of the sensor
repeat with a dry tissue
take picture and check for dirt
repeat as required
I use a 15 year old bottle of lens cleaning solution that the label wore off from years ago so I don't think it really matters what cleaner you use. Though, I'm sure some are a bit better than others they all work.