First off, I have a lot of respect for all controllers and don't envy you a bit. It's not a job that I think I'd be able to do. The vast majority of my experiences with ATC, and especially ARTCC have been very positive. With that said, here we go.....
1. I know this doesn't really apply to ARTCC but: Don't talk to an airplane while it's in the takeoff/landing phase below 1000 AGL, especially jets/turboprops, unless absolutely necessary. This is a very busy time in these airplanes. I'm particularly talking about frequency changes after takeoff....if you can put that off until the airplane has been in the air for a minute after takeoff, I think we'd really appreciate it. Of course if it's something critical, like "go around", "TURN IMMEDIATELY TO AVOID TRAFFIC" or "cancel takeoff clearance", please don't hold back!
2. Most airplanes can EITHER slow down OR go down, and EITHER speed up OR climb faster. Asking us to do both at the same time often just isn't possible, especially when a crossing restriction is involved. For example, let's say I'm cruising at FL370 and I'm given a clearance to cross a fix 100 NM ahead at 10,000. I'm not going to start my descent until about 70 NM from that fix, assuming zero wind, in the interest of fuel efficiency and best speed. If I get a restriction of "reduce speed to 250" as I pass through FL300, I'm not going to be able to make that crossing restriction. This is because I have to decrease my rate of descent in order to decrease my airspeed.
3. Don't get mad at us when you think we "missed a call". You probably said the call sign wrong. I know it sounds harsh, but I've experienced a lot of controllers getting "huffy" lately when they miss my call sign by a number or two and no one answers. When your company has up to 200 airplanes in the air at any one time, you tend not to respond unless the call sign is correct. I have no problem with the mistake being made, and the airlines share a lot of blame for making the flight numbers so similar, it's just the attitude of "AIRLINE XXXX, HOW DO YOU HEAR THIS TRANSMITTER???!?!?!?" on only the second attempt that gets old. If no one answers your call, look at the strip again to make sure you have the call sign correct. If after 3 or 4 calls for the correct call sign no one answers, then, ok, the attitude is warranted. With that said I often have to call a facility 3 or 4 times before I get an answer and manage not to get upset about it. Granted, check-ins are usually not time sensitive in nature.
4. Ride reports are ALWAYS appreciated by those of us hauling passengers.
5. If you don't allow me to deviate around a thunderstorm or towering cumulus, I may have to do it anyway, so be ready.
6. Shortcuts are almost always appreciated
7. If you need me to do something specific, just say so, don't assume that we'll operate that way.
8. Limit transmissions to no more than 3 instructions if possible. Anything more than that and I need an "advise when ready to copy" warning.
9. I'll always go out of my way to help ATC but if you're nice and cheerful, I think you'll get better results.
That's all I can think of for now. Just being friendly goes a long way.