Rimless toilets, and “not suitable for installation with flush valves”

I picked up some CLR and scrubbed all the jets under the rim. I even ran some through the tank - open flapper so as to minimize contact with the rubber and plastic. Using a mirror and a small piece of wire I was able to clean out a few stubborn holes.

All toilets were able to take a steady volume of water from a bucket without issue. This lead me to do a little more research. What I found was the hole at the bottom of the toilet that forces water down the drain is actually more susceptible to clogging and plays a larger role in the syphoning action. All three toilets had significant mineral build-up in this hole. A few minutes with my finger was able to clear out quite a lot. Some careful work with a small l-shaped alan wrench brought me the rest of the way home. Whammy!

The two younger toilets now flush like a champ. The oldest was so badly plugged I'll be needing a replacement. I'm gonna try a little soaking with CLR to see if it'll free up but I'm expecting a trip to the hardware store.

The Environmental Protection Agency has standards in place for a program that is meant to help conserve water. The current standards for "performance" of new toilets in this program (pdf) are as follows:

4.0 Flush Performance Criteria 

4.1 Toilet model performance is identified as either a Pass or Fail depending upon 
whether it can successfully and completely clear all test media from the fixture in 
a single flush in at least four of five attempts. Only toilet models that Pass qualify 
for EPA’s WaterSense label. Flush performance testing shall be conducted in 
accordance with the test protocol provided in Appendix A. 

4.2 Test media consists of seven test specimens, 50 ± 4 grams each, consisting of 
soybean paste forming a ‘sausage’ approximately 4 ± 0.5 inch (100 ± 13 mm) in 
length and 1 ± 0.25 inch (25 ± 6 mm) in diameter and four loosely crumbled balls 
of toilet paper as defined in Appendix A. 

4.3 The flush performance criteria apply to single flush toilets, and to the full flush 
option of dual flush toilets. No solid waste removal requirement applies to the 
reduced flush option on dual flush toilets. 

If your "media" is smaller than what's specified in 4.2, you should be fine.

When the US first started restricting the amount of water permissible per flush, the toilet manufacturers did not get the engineering right. So, if you've been using one of these first generation water conserving toilets, it could be the toilet that's the problem. The newer toilets have been engineered with larger diameter traps and coatings and what have you that make flushing "media" less of an issue.