US Computer Problems

We brought our desktop computer with us with the understanding that if we switched it to 230V, it should run fine.  Well, the first thing that happened was that my DH didn't know about the switch and plugged it in.  There was a loud pop and needless to say, the computer was dead. 
We took it in and had the power supply replaced.  They tested it and it worked fine.  When we tried it at home (making sure the switch was set to 230V this time), the light came on, it sounded like it was powering up, and then it went dead.  We tried again and the light flickered on for a brief moment and then went out.  Anyone have any suggestions?  Is our computer just not going to work over here?  Do we need to get a transformer?

A transformer isn't going to help you now.  I don't think replacing the power supply was enough.  I know you say the shop tested it and it worked fine, but I really think the motherboard is dead.  The memory is probably fried too.  It's weird that the computer worked at the repair shop though.  Were you there when they tested it?  Did you see it start up?  Were you given the opportunity to see if all your documents and such were intact on the hard drive? 
I find it remarkable that the motherboard would've survived such a power surge.   Perhaps it had just enough juice to run at the shop and died when you got it home or something.  It really doesn't take much to fry a motherboard.  I once ruined a motherboard just from a tiny spark of static electricity from my finger while upgrading the memory in my pc.

You might be better off just getting a new computer and having your old hard drive transferred to the new one, but don't do that until that repair shop has properly looked into your problem.

You could check inside the case to make sure everything is still connected. There are little wires that go from your case (usually at the front) to the motherboard - these could have come loose during the transport back home. They control the starter switch, speaker, IDE drive lights, power LED, etc. You may need to dig out your manual for proper setup if they are disconnected.

It's possible that connecting to 240V while set for 120V did indeed just affect the power supply and that this current failure is just a coincidence due to loose connections, etc.   Whether the power supply let through enough power on the higher voltage to damage the motherboard or other internal components depends very much upon its design. 

As suggested, as first step, just check inside that all the connectors are secure.  Equalize any static charges on your body by grabbing hold of the metal case with your hands before you touch the inside.

I'd second the two schools of thought mentioned above, looking first for a loose wire or connector, but fearing the second of a fried motherboard.  It could very well be new computer time.  You should be able to pull the hard drive and keep your data, but it doesn't sound good.

Take it back in to the repair shop and ask them to give it a look over.