EDIT: Never mind, you seem to have come up with more or less the same solution on your own while I was working on this. My version's a little more efficient in terms of space, but yours is easier to understand. You might want to look at my version anyway, if only because I used a few techniques you might want to learn.
Probably the easiest solution is to make a set of functions, each on of which prints a given die result, and call them from a jump table to print out the desired result:
def printOne():
print """
=========
| |
| O |
| |
=========
"""
def printTwo():
print """
=========
| O |
| |
| O |
=========
"""
def printThree():
print """
=========
| O |
| O |
| O |
=========
"""
def printFour():
print """
=========
| O O |
| |
| O O |
=========
"""
def printFive():
print """
=========
| O O |
| O |
| O O |
=========
"""
def printSix():
print """
=========
| O O |
| O O |
| O O |
=========
"""
# This is a random dice game.
import random
die1 = 0
die2 = 0
DIE_ROLL = [printOne, printTwo, printThree, printFour, printFive, printSix]
def dice_roller():
print "Dice Roller"
print "==============="
print "1) Enter 1 to roll a six-sided dice"
sum = raw_input("> ")
if sum == '1':
die1 = random.randrange(1, 7)
die2 = random.randrange(1, 7)
print "The first 6-Sided Die Rolled", die1
printDie = DIE_ROLL[die1 - 1]
printDie()
print "The second 6-Sided Die Rolled", die2
printDie = DIE_ROLL[die2 - 1]
printDie()
input = 'C'
while 1:
if input == 'Q': break
elif input == 'C': dice_roller()
input = raw_input("press c to continue or q to quit: ").upper()