If your mons have Protect, you can stall out 3 of the four turns easily by rotating between them- they each have their own protect timer. Prankster Substitute can also help stall out the timer. If you plan on stalling TR teams, having Toxic, Burn, Leech Seed, and/or Wish support can be helpful. You can also hamper Trick Room teams with well placed Taunts, Encores (assuming no Mental Herb), or your own Trick Room. They also have a hard time against defensive mons in general, since they don't care about speed, and a strong Dark or Ghost move can threaten common Trick Room users except Porygon2. Phazing moves will go before Trick Room (unless it's Whimsicott) and send the sweeper back, but these have little use outside of countering Trick Room and fail to work once you get your first kill. Hammer Arm users like Metagross and Conkeldurr can slow themselves down and try to outslow the Trick Room abusers. Spore, Sleep Powder, and Dark Void will always go before Trick Room and disable the user unless they have a Lum Berry or a Chesto Berry, or are already statused. Dual Screens help you weather the Trick Roomer's attacks. Priority attacks ignore Trick Room's effects (unless they use a move on the same priority bracket), but TR sweepers are generally bulky enough to take a few weak hits.
Most important to have however, is good prediction skills. Even without the proper counters smart rotating to resistant mons can waste the opponent's turns, and most of the counters I outlined are worthless if they hit the wrong opponent.