As originally stated, you should be able to mount your old drive a secondary (slave) drive on the new machine. If it isn't visible at all (i.e. no listing in /dev on linux - I guess device manager on windows??) and you really need to recover your data then I'd recommend getting yourself a copy of spinrite from **Home of Gibson Research Corporation**
It costs $89, but to be honest it is worth every bit - you only need to recover a couple of drives before it pays for itself - and if you use it properly for maintenance then you'll never have to recover a drive again!
FYI - many
studies have been done on new drives, and the results show that in the pursuit for ever-increasing density per platter and maximum platters per drive, has left the reliability of HDDs (and Multi-level-cell (MLC) SSDs in fact for those interested) in a dire situation.
New drives are being shipped with numerous sector errors already present, only a full scan through the drive attempting to read
the data (i.e. something like spinrite, rather than dban which only erases (overwrites
)) will force the disk controller to swap out bad sectors and resolve the condition. Not only that, but new drives are shipped with fewer 'spare' sectors so you won't have as many recoveries from sector errors under normal circumstances.
In summary - new does not equal perfect.