Efficient Sum of Primes
You've written a very literal and straightforward implementation to solve the challenge: test numbers for primality, and accumulate them until you have 1000 of them.

However, when you want to find many prime numbers, an algorithm such as the sieve-of-eratosthenes is more efficient than repeated trial division.  Here is a benchmark that illustrates the difference.  Your program follows the same basic outline as @Legato's solution, except that you haven't implemented some optimization hacks.  If you just want to sum the first 1000 primes, there isn't much of a difference in execution time, but your program will not scale up as gracefully as a sieve algorithm.