What is the best way to filter Turkish coffee?

I think you got something mixed up there. It is normal to have noticeable particles while drinking Turkish coffee. Everybody uses a strainer for it, and it does result in a slightly "muddy" drink.

I guess you could pass the prepared Turkish coffee through a paper coffee filter before drinking, but this would be a very unusual way to do it. People who don't want to taste the particles just use a different brewing method in the first place. Making it with particles and then removing them is twice the work, and I can't imagine it giving you any taste advantage.

If you do any filtering, the coffee will end up less strong, both in taste and caffeine. It is the particles you taste which are the actual coffee. Another method which achieves strong coffee is true espresso. It needs an expensive machine which passes the water vapor through the grounds under pressure though.

After you have finished brewing the Turkish coffee let it sit for a few minutes. Pour slowly into your mug stopping when you see the coffee turn cloudy. Immediately add sugar/milk to your own mug and stir. Don't stir again. The coffee grounds will settle to the bottom and become a clay like consistency. This way you will get very little ground in the coffee you are drinking.

I made turkish coffee with noticeable particles when I was about 7 yo, due to me not knowing that I should remove it from the heat only when the foam around the edge starts moving to the center, pretty much like this:

So are you sure you are boiling it enough time?

What is the best practical way to filter the Turkish coffee?

You won't need a strainer. Just pour it in the cup and let it be (DON'T STIR). By the time the temperature is about right to take your first sip, the coffee powder will rest at the bottom and the rest should be clear enough to not bother your tongue papillae. This happens probably because when the coffee is cooked well, it gets heavier. Stop sipping when you have reached the 'muddy' bottom, and the powder starts to get in your mouth.

Other tips:

You should mix it well at the beginning, but NOT further. This helps in many ways:

1. You will be able to see when it is ready with the 'foaming around the edge' method I mentioned above. When it's heated, it forms a light-colored foamy-like crust that puffs near the walls of the coffee pot and moves to the center to indicate that you should remove it from heat. If you are stirring it constantly, none or little foam will be formed.
2. This crust is called kaymak. In order to have kaymak in your coffee, you should pour it in your cup carefully - from a small distance and slowly - in order not to mix it, because it will disappear. Some say that coffee with kaymak is tastier/stronger. The ideal display of your kaymak should be like this:

It should have little bubbles and some holes that make the coffee beneath visible. If it is solid like this:

It means you put too much coffee/didn't boil enough time/let too much kaymak slip and you will experience that unpleasant - in your opinion - sensation of coffee particles in your mouth.

Good luck with your next try!

I use a very fine cone filter. Takes slightly longer to filter. It does get rid of most of the grinds and then some, and while it is a bit weaker, it's still stronger and tastier than regular (drip, melita, etc, coffee).