All these recommendations are good and will provide the fastest way to access your data on the old drive but at up to $30, if you've only one drive to do, it might be a tad expensive. If you have more than one drive this is the way to go.
But there is an alternative that is cheaper, depending on:
1. Is the drive in question still in the old computer?
2. If so is the old computer still working?
3. If so, does it have an Ethernet port?
If all three answers are "yes", you could plug an Ethernet crossover cable between the old machine and the new one and, hey presto, you have a peer to peer network and can access the old drive from the new machine as a network drive. If you have a router, it's even easier.
The old machine is unlikely to be better than 100baseT, so it won't be very fast but for just one drive you can likely put up with that.