Some of this has been said, but I'll give my take on it after playing with the new orks for a while and against them a few times.
There are 3 things that Orks don't do well and that a crafty opponent can capitalize on (I'm only mentioning viable ork units here. Things like flash gitz are a little unreliable):
1. Taking out medium to heavy vehicles (AV 12-14), especially heavy vehicles. Orks can field a lot of good str 5 weaponry and so can deal with light vehicles very well but their str 8 weapons are few and generally inaccurate and they've really not got anything higher than str 8. Rokkits are too inaccurate in ork mobs and tankbustas are a point sink that have to have a vehicle purchased for them, so the most effective str 8 weaponry they can field are warbuggies, deffkoptas, kannons, and killa kans which are all fairly fragile. Lootas are another ork option to deal with light to medium armor, and unlike the other options actually have good range, but they too are fairly vulnerable, even in 4+ cover.
2. Taking out heavily armored infantry from range. The only thing that the orks have that's good at taking out heavily armored foes (terminators or other 2+ saves) is the shokk attack gun. Taking it out is very difficult due to the mek's independent character status. If your opponent lacks one of these, you're good to go, provided you can stay out of melee range. 3+ saves are also somewhat difficult for orks to deal with, but the incredible ork massed firepower can still reliably bring down marine squads. In melee, powerklaws generally make up for the orks not dealing with 2+ saves at range very well, so keep your terminators out of melee with orks (shocking, i know).
3. Taking out foes at long range. Orks aren't much of a threat until they're up close. They aren't going to be much of a threat on the first turn, or even second sometimes if you've taken out the right units. Most of their weaponry is limited in range. 18" away from orks is where you'll get hurt the worst. 24" is the max range of many of their special stuff. 36" away all you've really got to worry about are big shootas and maybe some ordnance. The exception to this is again lootas with their 48" range. Seeing as how they do well 1.5 out of 3 things the orks do poorly, they should be a priority target.
Ork armies are very competitive now, but a lot of fun to play and play against. There are 3 basic flavors: mechanized, horde, and horde with fast elements. There are also deffwing armies and dreadmobs, but those are rare and not that effective. Biker armies are a little more common (and I play them fairly often) but very vulnerable to getting shot up.
Horde - The really helpful thing to do against horde orks is to deploy yourself with a denied flank (basically you put the bulk of your army on one side of the table while his size forces him to spread across the whole table, thereby forcing half of his army to have to run an extra turn or two to come to bear on your forces), which works against all horde armies. Focus your efforts on one portion of his army at a time and you'll not have much trouble, provided you're not specialized for anti-MEQ (thousand sons armies just don't handle orks well).
Horde with fast elements - This army is IMO the most lethal ork army. The army functions as anvil and hammer, with bikers or stormboyz or other fast units hitting one flank hard on the ork player's second turn or taking out the good anti-horde elements of the opponent's army. The horde elements then follow up, mopping up anything remaining. At worst, the fast elements buy the horde elements a few turns of walking to get within their effective range. Usually it forces the opponent to try to engage both elements at once, allowing both to do their job. The best part about this army is that it isn't affected by the denied flank deployment strategy that works so well against hordes. Simply being able to bring your whole army to bear against your foe is worth it in and of itself.
Mechanized - Mechanized ork lists are very common, but there is one reason that I won't use them that most people seem to forget. Sure ramshackle protects the boyz if the trukk gets destroyed, but if someone gets a penetrating hit on your trukk and doesn't destroy it, there's still a forced disembarkation, and 5 of the 12 orks aboard die, on average, neutering the squad. The way to fight mechanized orks is to target the transports first and then go after the squads within.
Whenever I fight orks, I simply prioritize my targets and eliminate threats as they come up. For example, with my tau against a mechanized ork army with 2 big loota squads I deployed my tanks on the right half of the table, leading my opponent to put much of his army on that half, and then I put my army on the other half, resulting in a large part of his army taking an extra turn to get at my troops. First turn I wiped all 20 lootas out of their 4+ cover with my shooting. Then he moved his transports (which were closer to me) and his bikes (which were further away) as far as possible. Then I shot up his transports and the squads that they carried. Then his bikes came up, so in my turn I wiped them out and worked on some of the entangled ork passengers. Against horde armies you can just dwindle the mobs as they get closer (focusing on fast elements first), and then focusing on any that get too near.
As orks, there are certain things that really just kick my butt for various reasons. War walkers with scatter lasers x2 (and guided by a farseer) outrange my stuff and utterly rape pretty much everything orks have. Hellhounds scare me too, though I haven't had to fight one yet. Getting flamed from 24" away by a str 6 ap 4 flamer is scary, and orks have a hard time with armor. Space Wolf terminators with runic charms earn a place in my list for each having an extra wound, even if you get through the 2+ save. If you're shooting at a unit of 6, and manage to get 6 failed saves, you've maybe killed 1 terminator (1 in 6 will fail their reroll). The eldar doom power is pretty harsh too, since orks have to get close. Bladestorming dire avengers into a doomed ork mob are devastating, and the fact that doom allows multiple eldar to shoot at the same mob with the reroll benefit means that big ork mobs get taken down efficiently.
Other templates are OK but if the ork player spaces out his boyz then each shot if going to kill only 30 points worth of orks or so out of normal mobs. Any anti-infantry weapons with lots of shots are generally good though.
Hope that helps. If I think of anything else I'll post it.