Multiplication tables can be the bane of a second grader's existence, but they don't have to be. The 3 times table, especially, can become second nature with the right tricks. The number 3 is a building block of many other numbers, and therefore knowing the 3 times table will help you simplify fractions containing multiples of 3. Before you know it, you'll be able to recite the multiples of 3 without even thinking.
The easy ones

The first two lines of the 3 times table are very simple, and follow the same rules as the rest of the times table.
Any number multiplied by 0 equals 0. So, 3 x 0 = 0.
Any number multiplied by 1 is itself. So, 3 x 1 = 3.
Two other lines also follow universal rules:
To multiply any number by 10, add a 0 to the end of the number. So, 3 x 10 = 30.
For any number under 10, you can form the multiple of 11 by simply repeating the digit twice. So, 3 x 11 = 33.
The order in which you multiply numbers does not matter. In other words, 3 x 5 is the same thing as 5 x 3. So if you already know your 2 times table, then you already know one line of the 3 times table (3 x 2 = 6).
Making it automatic...

Multiplication is repeated addition. In other words, to multiply 3 x 4, you are actually adding 3 together 4 times: 3 + 3 + 3 + 3. (see multiplication.com/lesson1_repeated_addition.htm)
It can help to visualize the times table. Use marbles, peanuts, pennies, or anything else small and plentiful. Make 12 groups of three, and then count. In the first group of 3, you obviously have 3 items. This reminds you that 3 x 1 = 3. When you count both the first and second groups, you have 6 items. This teaches you that 3 x 2 = 6.
Continue counting, each time adding one more group. You've just illustrated the 3 times table.
The key to memorizing your multiplication tables really is repetition.
Write your 3 times table several times throughout the day. If you're learning this as part of a group or class, recite the times table out loud, together.
(Read this lesson plan from a school in Oregon for description of activities to facilitate repetition: coled.org/cur/math/math02.txt)
Quiz yourself, or ask others to quiz you. First learn the times table in order, and once you feel comfortable with that, quiz yourself out of order.
Tricks of the trade

Numbers seem like magic sometimes; they just fit together and form patterns in ways that make everything easier. This is the case for the 3 times table.
Take note of this rule:
In any multiple of 3, the digits will add up to a multiple of 3.For example:
3 x 6 = 18.
The digits of 18 are 1 and 8.
1 + 8 = 9
9 is a multiple of 3. (3 x 3 = 9)Another example:
3 x 10 = 30
The digits of 30 are 3 and 0.
3 + 0 = 3.
3 is a multiple of 3. (3 x 1 = 3)This rule can help you check your answers once you've calculated a multiple of 3. The digits will always add up to a relatively low multiple of 3, so you don't need to know the entire times table to use this checking system.
In sum, the 3 times table won't feel intimidating as soon as you visualize what it means and begin ingraining it in your mind through regular repetition.