Open Picker window, find the rough section of speaker type/cab/size you're after, pick the first one. Since all the IRs in the III are laid out together in groups, I then select the "#" box in Axe-Edit Cab block UI (so it has the little dashed box around it, just click the # sign to the left, not the text entry field), and then with that # box selected you can now use the down or up arrow keys on the keyboard to step through the rest of the IRs around the one you initially selected to get a different mic... so then it's just riff riff riff, arrow key, riff riff riff, arrow key, riff riff riff, arrow key... I audition all the mic types for that speaker/cab in this way until I find a response/sound I like, lather, rinse, repeat.
Alternately you can click the little pin button in the upper right corner of the Picker window and that will keep it on screen as you click with your mouse, you could then just click your way through all the cabs, but I personally find moving picking hand from guitar to mouse to move down slightly to click the next IR a bit more interruptive to the flow, so that's why I prefer the "arrow key on the number box method" as it's really quick for my picking hand to reach up and smack the arrow key, it's less precise of a target for me I guess, but YMMV. You can also arrow key up/down through the picker UI, but it does require hitting the enter key then once you have the IR name highlighted to actually load it (can be faster/easier than messing with the mouse if you want to cover a bigger distance in-between IRs, but if you're just stepping down/up through all the mic's of a given cab the extra enter key press is just another unnecessary step over doing the down/up keys with the # box selected, but I do that sometimes over reaching for the mouse/track pad).
Another option would be to put a looper block at the start of the patch, play a quick riff, let it loop, and then man the picker window and change IRs w/o having to have hands on the guitar strings.
