Orphaned, partitioned hard drives - will I be able to read them?

Do not worry IF you are using the right term. ALL HDD's are Partitioned.

A new HDD is just empty space with no organization. Before it can be used by any OS, two steps have to be done. First, one Partitions the HDD - that is, you use a utility to define a chunk of that empty space to be used as one "drive" with its own letter name. You can make one Partition to use up the entire HDD, or to use less than that. If you don't use it all up, you can define a second Partition in the remaining space and it will become a separate "drive" with its own letter name, even though it is physically located on the same HDD unit.

After the Partition is created, you then Format that "drive". This operation installs on that Partition ("drive") all the hidden files required to read and write data and track all the files you make. Once this step is complete , then your OS can recognize and use the drive.

If you create more than one Partition on a single HDD unit (and this is what OP says is the case for the old units), what you will see in My Computer is two (or more) "drives" with their own letter names. You will NOT see them as one "drive" with special "Partitions".

On some computers the manufacturers have placed a special hidden small Partition on the HDD that contains a set of backup files for the OS installed in the main Partition. If something gets corrupted, there is a way to replace the corrupted files with copies from the hidden partition. But you never use that hidden partition for anything else.

OP, how you proceed depends on what your longer-term plans are. If you want to mount those old IDE units in your case permanently and keep using them, you'd need IDE ports on the mobo (or via an IDE controller card).

If you want to use them as externals, but keep using them often, you will need to buy one or more External Enclosures. When you buy, there are two key pieces on info you must pay attention to. First, what is the enclosure's INTERNAL connection system: IDE or SATA? You can get either. Secondly, what it is EXTERNAL connection system - how does it plug into your computer? Common options are USB2, USB3, eSATA, IEEE1394a (Firewire 400), and IEEE1394b (Firewire 800). Choose according to what ports your computer has to use. Many enclosures come with more than one connection option, but you only get to use one at a time. Mine has both USB2 and eSATA, and I use eSATA because it's faster.

On the other hand, if what you want is really just to connect the old drives, one at a time, to your computer to copy old files off, and then plan to simply store the old drive units, you could buy an adapter that lets you plug a bare IDE drive into a USB2 port, with power provided to the drive. These units often are cheaper than an enclosure, and it is easy to switch from one drive to another. For example; this:

http://www.newegg.com/Product/Product.aspx?Item=9SIA1PU0GG9972&cm_re=ide_to_usb_adapter-_-9SIA1PU0GG9972-_-Product

can be used for either IDE or SATA HDD units. Simpler (and cheaper) ones exist for IDE drives only.