Step 1: Realize that during a drought you will lose some plants. If you're going to water to save plants, make sure you're watering to save the correct plants. Note: Your lawn should never be the plant you opt to save during a drought. Your shade trees, on the other hand, should be.
Step 2: Obtain yourself some "tree waterin' bukkits." I am partial to using old cat litter containers. Since I've switched to the type that comes in a giant jug, I can use them for everything! Tree watering buckets, compost tea mixing buckets, seaweed and fish emulsion dilution, etc. Wonderful things, they are. Tip: You get extra credit for making these as 'white trash' as you want. I highly recommend writing "tree waterin' bukkit" on them. But you can do what you want.
Step 3: Figure out how much water your tree watering buckets hold. I used an old milk carton to measure water. Each of my green jugs holds 2.5 gallons.
Step 4: Drill some small holes in the bottom of your bucket. I didn't even bother getting out the drill - I just grabbed a drill bit (probably the 1/16th bit) and pressed it into the bottom while twisting until a hole formed. Each bucket has 3 holes. It took all of 5 minutes to do both buckets.
Step 5: Calculate how much water each of your trees needs.* Now this is going to require you to do a bit of math (or maths as our British friends say) (Note: I'm using Linux so it always tells me that 'math' is incorrect because apparently the Fedora OS is British). Firstly, figure out the diameter of your tree by measuring the circumference of the trunk at about chest height. Reminder: circumference means 'around.' Once you have the circumference (in inches), divide that number by 3.14 (Circumference = pi*diameter). Voila! you have your diameter. Use the handy table below to figure out how much water you'll need for your tree. NOTE: If the tree is stressed, use an amount at the upper end of the range, if the tree is healthy you can use an amount at the lower end of the range. NOTE2: Trees planted within the last year should always get 10 gallons, and they should be watered closer to the trunk in step 7 since all the roots are going to be closer to the trunk due to being still roughly in pot shape)
Step 6: Figure out how many bucketfuls of water you'll need to use on your tree - you can accomplish this by dividing the number of gallons needed by the number of gallons in your bucket - that's why I have two buckets - two buckets = 5 gallons. Much easier to divide by 5 than by 2.5.
Step 7: Line up your bucket with the dripline of the tree - that is roughly the location where the branches end or further out - no need to water right next to the trunk. Fill your bucket with water and come back 15-20 minutes later as the bucket will take a while before it's drained (that's the point - low and slow...just like barbeque!)
Step 9: Voila! Your trees are watered and now they won't die. You're welcome.
* Iffin you don't like my wording or directions, you can see an official City of Austin handout on tree care during droughts here: http://www.austintexas.gov/sites/default/files/files/Watershed/growgreen/treecare.pdf