In my opinion as the last guy stated it is much easier to go from West to east. I will try to write it out so that the google maps etc will be easy for input.
From where ever you start in the valley make it up to Cave Creek road. You want to head east on Cave Creek road.
Eventually as you go east you will find the intersection of Cave Creek and Bartlett Dam Road (Also known as service road 205) continue on straight ahead on Cave Creek. After you go through several really nice twisties the road will turn to dirt.
At about where it turns to dirt the road is renamed 7 Springs road. Keep on trucking this road continues through the 7 springs rec areas and campground areas. There are 2 nice water crossings. Eventually you will also see forest signs that call this road FR24. (My street car GPS even calls it this) Keep on this road for about a dozen or more miles. It will eventually come to a T intersection. The road to the west goes to(Left) Bloody Basin and the I-17 freeway near Cordes Junction. The East (Right) goes to Sheep Bridge. Both ways are called FR269. Make the right and head east to sheep bridge. Once at Sheep Bridge carefully ride your bike across the bridge. It is a bit narrow but not bad just go slow. Once across the bridge there is a fairly nasty downhill section the leads to the beach area. I recommend that you coast down the hill with your engine shut off. This will prevent whiskey throttle and near death experiences. I am sure that there are plenty of guys that can ride it down but at least on the first try it might be smart to take it easy. Once you get to the bottom of the hill you want to find the road that continues to your left or east that goes to the dam. About 5 miles or so down this road you will see the dam on your right and continue at least a mile or two past where it is visible. There will be a fork in the road and a sign that says (someone's) farm. Take a right at this sign and not too far up you will find the dam. (If you find yourself crossing alot of water and in a river bed you missed the turn. Been there done that.) Once you cross the dam under on that pedestrian crossing there is a nice steep and pretty hairy part from the ped xing to the road. You might want to walk it up. If you slide it is a long way down. Once on top the road from there goes several miles of really crappy road to a T intersection. At the T make a right onto the Bartlett Dam road. You will ride a short ways and there will be another T intersection. This is the intersection of Bartlett and Cave Creek. If you make a left here you will be heading toward Carefree and Cave Creek. Closest gas will be 3 miles ahead on Cave Creek at the Carefree intersection.
I recommend that you ride a light bike. I think that I could possibly do this on my VStrom if my life depended on it, but it wouldn't be fun. The KLR would be okay too but it was really fun on a dirt bike. The KLR was doable just not really alot of fun. I had to work too much. When I rode it on the XR350 it was a really fun day. I think the guys on the DR650s had a good time and Jack rode his KLR and enjoyed it but he is a pretty big guy. In any case watch out for the rocks when you head up the hill leaving the dam crossing it will leave a wet spot in your shorts. Good luck to you. -Paul
P.S. Anytime that the weather is nice. (Less than 100 and not real wet) would be a great time for this ride. It tends to be windier and cooler so it helps make for a nicer ride with gear on. I know that my fat butt gets overheated with all of the gear, but on this ride I was comfortable all day.