It depends - if the ground has frozen solid for the season (as it does up north) then you wouldn't water at all, just let snow pile up and melt in on warmer days, but in Cal. I don't think that would apply, so it's really a matter of spacing out the watering times a lot more - don't necessarily allow a plant to totally dry out for any length of time, but water less often than otherwise. However, you don't say if you're talking about something in the ground outdoors, in which case does it not ever rain there (what would any plant do?), or if it's indoors, dormancy is kind of relative - a plant will stop growing actively for a few weeks only, if that - so you just have to be careful about not soaking it all the time the way you would in spring (when most plants can take a lot more).

Once the trees go dormant here and the weather remains cool, I only water once a month if it doesn't rain at least 1". Dormant trees in pots are watered about once a week if no rain.

Scotjute- Would you apply your schedule to recently set out trees with large root balls?

Also, what temperatures do you experience during the dormant season?

Depends on the type of soil, the weather conditions, and the kind of tree. Check your soil. Most trees prefer well drained but moist soil. If your tree is like most trees, water as necessary to ensure the soil at a depth of a couple of inches doesn't dry out. There are too many variables for a simple quantitative answer. Observation, experience, and common sense are your best guides. Established trees planted in the ground should only need additional watering if there is a drought.

hoseman,
What I gave is a general rule. brandon7 has good advice. The answer is yes and no.
I just put out 2 Bur Oaks, 2 Black Cherry, 2 Live Oaks, and 1 Hinoki Cypress immediately after a 2" rain last weekend.
Everything is going dormant, I know this because leaves are falling. The water in the soil will last for at least a month with our cool temperatures, by then we will surely have had a freeze and deciduous trees will all be totally dormant and I will not have to water! Yippee!
That said, I will probably water these trees about 2 more times in the next week unless we get more rain. I do this to help insure soil compaction and elimination of air holes around the ball of the tree, not because the tree actually needs more moisture. (soil is clayey).
Temperatures down here are pretty variable during winter. We can be freezing one day, and 80 F the next. Normal temp.s during dead of winter (Jan.) might be in upper 30's with daytime warming to around 50 F. We usually have several freezes in the low 20's F each year.

Scotjute,

You are safe till next summer. No more heavy watering needed. in fact, overwatering is #1 killer for newly transplanted. You have clay soil and it will hold water for a LONG TIME at the time of the year. If anything I'd put down only one or two gallon of water directly into rootball which tend to be pine bark mix once or twice a week. It dries out fast.

ALL trees needs at least 2 inches of rain or watering once a month. Sudden downpour doesn't count since the water will just run off unless it's a very flat land. Very slow soaking like soaker hose works best. We don't always get very slow soaking rain unless the ground is much improved covered in mulch where they act like sponge soaking up all the water quickly and slowly release water deeper into the soil. That's why a lot of large trees look bad where there are bare ground and water just run over them during heavy rain.