How do I clean my jetted whirlpool tub?
Were the previous owners kind enough to leave the instruction manual for the tub?  If not, you may be able to look up the model online, and get a copy.

The last jetted tub I had experience with, had removable jets, which could be placed in the dishwasher (as per the instruction manual).  In order to clean out the tubing, we would fill the tub, and add 1-1.5 cups of bleach, then run the jets for a good 20-30 minutes, drain the tub, refill with clean water, and run for another 5-10 minutes.  It was amazing how much "ick" came out, and with the bleach you don't have to worry about buildup.

If the previous homeowners were as you say, lacking hygiene, you may want to repeat this process a few times.