I understand why a hashing algorithm should be slow but is the method that makes it slow important to the strength of the hash? Everything I've read says that the algorithm should be computationally slow - hash the thing over thousands of iterations or concatenating it with huge strings to slow it down. This seems like it would put unnecessary strain on the CPU. Couldn't you just hash the password once with a good random salt and then just pause the thread for a set amount of time?
The goal isn't to make the hash slow for you to compute. The goal is to make the hash slow for an attacker to compute. An attacker with fast hardware and a copy of the hash and salt, giving him the ability to mount an offline attack, to be specific. The attacker need not pause a thread during his computations just because you added that to your software. He is going to compute the hashes as quickly and efficiently as he possibly can. Therefore, in order to make it computationally hard for him, with all of his fast hardware and his efficient hashing software, the hash must be computationally hard for you to compute as well.
Running a few thousand iterations or appending a very long salt is not necessarily the best way of making a hash run slow, but it is the most obvious (easiest) one.
The intent is first and foremost to slow down massively parallel attacks. An attacker will not be able to brute force your hash in reasonable time on a single CPU. He will be using either a botnet or a machine with several GPUs (or both).
Running the same hash a thousand times is indeed not exactly the wisest thing at all, since it is easy to run this task in parallel, and it mostly stresses ALU, both being perfect matches for GPU computation. On the other hand, the impact on your CPU may be very noticeable (depending on how many iterations you do).
Ideally, a slow hash would involve operations that make parallelization hard and that utilize bandwidth in addition to raw ALU, such as requiring and touching substantial amounts of RAM (this also limits the amount of instances that can run on a zombie without being detected) or requiring operations that are expensive on GPUs (e.g. random access writes).
Of course this is not exactly as trivial to achieve as simply running a hash some thousand iterations, which is why one usually resorts to doing that.