Is filtering Japanese matcha green tea drink recommended?
Matcha is supposed to be thick and frothy - one of the reasons it is traditionally prepared with a whisk is to froth up the texture - and filtering it would remove both the texture and the powder that has taken considerable effort to get properly suspended in the tea.  

Matcha tea is very expensive because it is exceedingly high quality, and carefully ground smooth and stored well, and this is because the powder is all meant to be ingested.  If it was meant to be filtered out and removed, then it would simply be green tea, and it would be much easier to have larger flakes and lower standards.

Smaller particles will brew faster, that is pretty much the advantage of powder, but they will also overbrew very quickly and can introduce off flavors (compared to whole leaf tea) if the quality is not very good.  Teabags make this tradeoff, using crushed powder (fannings) which are meant to be steeped very quickly and removed, and usually end up being made of lower quality because it;s use is that steeps fast and strong, and the finer qualities of high quality teas are usually not preserved through the process.  Matcha makes a different tradeoff, if it is to be worth using only the highest quality leaf, the results must be much, much better than regular green tea for it to be worth the extra effort and expense - so the leaf is carefully selected and drunk whole for a very different experience than regular green tea, not strained out (which would give a teabag-like experience).