Well ... there is a way, but it's kinda technical. I'll explain and let you decide if this option is for you. :)
You can go buy an external hard drive enclosure (they usually sell for about $25) for your hard drive and then put the old hard drive into the enclosure. Yes, it does involve opening up the old computer and taking out the hard drive, but it also means not having to set it back up - ya need to pick your battles - haha!
This also works great when the old machine doesn't work well or has stopped working all together.
Next, you plug the external hard drive enclosure into your new PC and you can access your files - normally. The only issue occurs if you had a password on your profile, in which case you would need to do a little more work to get to those files (I oughta make a hub out of this - and I will!).
From there, you can transfer the fies to the new PC (for quicker access) or leave them on the old hard drive. Either way, at this point you have won!
I hope this helps. :)