As in, to the side either three or six inches? Is there any give to whatever sort of connections are on the back?
Assuming this is a built-in under counter dishwasher, there is not much for fastening. Usually two screws on brackets at the top into the underside of the counter, or though the side into the cabinetry. Open the door and take a look, screws at the top should be visible, ones in the side are usually behind little rubber plugs.
The connections for electrical and plumbing are often pretty tight so that could be a problem. Is there really space to move it to the side or are you changing some cabinetry?
It totally depends on how the dishwasher was installed in the first place. There are 3 connections you need to worry about – electricity, water and the drain line.Usually the water and drain are flexible hoses, so there should be some amount of slack. Probably the main thing that may limit you is how tight the electrical connection is, and if the installer left any slack. Even if he didn’t leave any, it should be possible to run a new wire either from the box or the nearest junction, but you may need an experienced electrician to do this if you’re not up to it yourself.
ETA: The connections are not on the back, they are underneath the dishwasher, in the space between the tub and the floor. You can remove the kick plate underneath the door of the dishwasher to look underneath with a flashlight and see what you’re dealing with.
I’m changing all my cabinets and have no idea what’s back there behind the dishwasher. If I can’t move it I’m going to have 9 inches of unusable space because of cabinet sizes.
ETA - I’ll have somebody installing it (either the cabinet hangers or the plumber) I just need to know if this is going to be a $50 job or a $700 job.
On rereading I see you probably mean more specifically the plumbing and electrical. The supply lines for water come in different sizes and if the existing one is not long enough you can get a longer one. The electrical may be more of a problem, often is is just a cable coming out of the floor or wall, and will require a junction box if you want to extend it.
If you are changing all the cabinets then no problem, get the cabinets you want and plumbing and electrical can be made to work. We do it all the time.
Agreed. It’s a simple matter to get a longer plumbing or electrical connection when everything is already exposed. You should be just fine.
As I measure for the 50th time I note that the best cabinet arrangement (the one that doesn’t involve moving 6 inches and therefore moving the plumbing for the sink) is VERY close to the sink plumbing connection. So close that theoretically it SHOULD fit but in reality there’s honestly no way to tell once it’s actually installed to a real life wall. Is this a dealbreaker?
I’m not 100% sure what you’re asking here, but whoever is working with you to design your cabinet layout will be able to answer that. From what I think you’re saying, you should be OK.
I’m going to Ikea, so while I’m sure they’re helpful I bear the final responsibility for this. 
I’m going from a two bowl sink to a one bowl sink, so the sink cabinet will be smaller. Everything is moving three inches to the left, so it will go - cabinet three inches shorter, dishwasher three inches moved over, sink cabinet much smaller and 3 inches over, big corner cabinet with pullout. The sink cabinet is shrinking AND moving, and plus I’m trying to measure plumbing connections in existing cabinets which it turns out is hard as hell to do and not very exact.
it would be a $50 to $100 vicinity job likely. get a plumber (or plumbing experienced) to do it, a bad installation can become expensive.
the water connections could be flexible hose and make this easy. the electrical connection could be flexible cable as well.
if you are moving the dishwasher to the side the hose is coming from (in the basement) then there is no need to replace the hose because you will have the extra hose there.
also open or closed cabinet slot for vertically stored cookie sheets and shallow pans would not leave that 9 inch space wasted.
this thought before messages 8 and 10 which i haven’t digested yet.
The plumbing should be trivial, you have to disconnect all the hoses anyway, so the change would be buying longer hoses when you reconnect. The electrical may require running a new wire from the closest junction box, putting in a new box behind the dishwasher to extend the line.
You think the edge of your sink cabinet may line up with one of your supply lines? If it does, you could cut away the cabinet where the line comes in. You may have to do some custom work on the cabinet to attach it to the wall, but it’s just screwing wood to a wall, it shouldn’t be outrageous. Or, once the old cabinets are out, break into the wall and re-run that section of pipe to a better spot. That should not be a terribly expensive job.
BTW, save your receipt for the cabinets. IKEA has a 10 year warranty, but it’s worthless without the receipt.
As I’m reading this, I am looking at my dishwasher which is flanked by a small cabinet with a six inch door. Ideal for storage of baking sheets, oven racks etc.
Perhaps something like that would work for you.
I’d love to have a baking sheet cabinet, but Ikea doesn’t have a door that size.
Most dishwashers I’ve seen just plug into an outlet. If not, you should get an electrician to put a cord on yours. There is no reason to hard-wire the dishwasher to a junction box.
Check your current dishwasher, and if you have to, just put a longer cord on it.
Just FYI, you don’t have to use Ikea doors on Ikea cabinets. Other companies make doors that fit them, and have more options. Scherr’s is one that I’ve heard very good things about, and supposedly can be cheaper than Ikea doors.
Could you get their narrowest cabinet and have it cut down to size? Might be less trouble and expense than moving the dishwasher.