not boring Old towns where pensioners hang out :)
In fact, some of the best places are up near the Old Town. Le Madame is sadly long gone, but in the bit of the Old Town that's actually called the New Town (don't ask), there are some fairly cool places if you know where to look.
You might like Bar Warszawa at the top of ul. Krakowskie Przedmiescie, corner ul. Miodowa (tourist central round there but don't let it put you off). the bit you want isn't the front part (though that's ok), it's the back bit from the courtyard off Miodowa, especially the upstairs bar.
Also, the bar on the ground floor of the Pałac Brzozowskich (down an alleyway off ul. Bracka (use the main entrance, you probably don't want the all-male establishment round the sides) and there are some places you might like in the alleys and courtyards off ul. Chmielna (the Nowy Swiat end, definitely not the other end).
Stara Praga (across the river) is another possibility, especially the bars and clubs off ul. 11 Listopada. That area's sort of becoming Warsaw's Kreuzberg.
Try Chinatown though - the best bars there are by the way often the ones that look as if they're closed.
Too late to edit - here's a site about Praga: klubynapradze.pl
And here's something about Chinatown: yelp.com/biz/pawilony-nowy-świat-warszawa