The problem with bash or unix-y tools is that if any of the filenames have spaces like Windows LOVES to do, you'll need to deal with that, either do those by hand or edit the script you create to properly escape the spaces, or whatever.
If they're in directories that Windows thinks it should be in control of you'll have to jump through more hoops. Assuming they're not, this is the way I'd do it:
Assumptions: There's a reasonable number of files - more than 10 or 15 (fewer than that and I'd just do it by hand) and fewer than about 100 (then creating the list below will be unbelievably tedious unless you already have the list in a file, and you'll have to get really clever with automation)
1) Create a list of the files and destination paths, one per line, and start each line with "COPY /Y". ie "COPY /Y afile.dat C:\Games\mygame\data\afile.dat". Name this file with all the COPY commands something like MYCOPY.BAT.
2) Transfer the files into a temporary directory on computer A (probably making an archive on computer B and tranferring just that to computer A). (If you're good with an editor, you can use the editor to reverse the fields in the file to make the script to copy each file from the path on computer B to a temporary directory in preparation for the transfer - I know how to do this in vi/gvim but nothing else, alas. A careful person would also use a slight modificaiton of the script to back up the existing files on computer A, just in case).
3) On computer A, start a command prompt, change directory to that directory you just transferred the files, run MYCOPY.BAT to copy each file from that directory to the correct path. Pray...