I may be doing a thru ride at the end of may
The end of May would be a good time to go from Payson to Utah or vis versa, below there you could do the dirt road route but the ST would be brutal in the heat.
In the late summer of 06 Troy, Blair and Brian"s did a North South trip
http://www.juniper-solutions.com/aztrek/ they had to bail off of night riding the remote ST on account of rattle snakes they ran over 6 in one night and found it so hot on the highway the next day they had to bum a ride. Not that they didn't have a good ride in the end just something to think about.
You could possibly do the low country roads if you only rode in the morning and evening but the midday heat will get you and it just doesn't seem good to be out in the back woods when there are lots of snakes out.