Non-weather is hard to build, and when you build it, consider threats from four (if hail is counted) weathers. There is keldeo in rain, the probability of garchomp and thunderus coming back from ubers (this is quite unlikely, considering that scarfchomp might be needed to deal with the new kyurem formes) and the therain forms everywhere. Perhaps an FWG core consisting of amoongus, heatran and slowbro might be able to weather through the battle.
We'll deal with Garchomp and Thundurus-I when the time comes, but there is no need to consider them right now.
And, weather really isn't very hard to counter, as long as you have the right strategies and pokemon. You're right- Amoongbro+ Heatran handle weather very well. Ferrocent+ Mamoswine are also great weather counters.
I agree with Scarfywynaut; not having weather is basically like a whole nother pokemon if you can handle it. Though, Ttar does work pretty well for Stall, since it eliminates Espeon quite well, who is a downright bitch to handle if you don't have Ttar or some other physical attackers.
I don't know if it counts as weather, but I've been using Trick Room as a back up strategy for my Stall team, and boy has it saved my ass- bringing me from a 6-3 in my opponent's favor to a 2-0 in mine. Trick Room is certainly a field effect many nonweather teams should be ready to face; especially because there's no stopping it once it's up.
EDIT: @ColonelM: Back when I used a full choice item'ed Rain team, I frequently sacked Politoed as death fodder because unlike Ttar it's relatively useless besides setting up rain. I also found that weather teams are also constantly at war with other weathers- many rain sweepers can't do shit in sun, while things like Venasaur or Sawsbuck just fail in Rain. They are so reliant on their weather, and that's what makes them easier to take down than most people think.
Also, am I the only person to slap in Hail on one of my pokemon just to screw with other weather teams? I don't use it anymore but boy is it fun seeing other people rage