Green slime--not just normal slimy debris? Can you post a photo? What type of filter is it? Filter foam does get mucked up normally--just squeeze the foam several times (in a bucket of your tank water when you do a water change) to remove lots of it. It doesn't have to be pristine--if fact, it shouldn't be too clean. I'm guessing this is what you are experiencing and it is totally normal.
If it is really, really slimy to the touch, there may be something else going on. Is this the tank you've had your bad luck with? If so, a massive die-off of beneficial bacteria will cause a clear slime to develop. A photo will help.