A sequence is defined as follows $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}=6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum^{28}_{k=1} a_k$ is divided by 1000.
1. We start with the given sequence definition:
   \[
   a_1 = a_2 = a_3 = 1
   \]
   and for all positive integers \( n \),
   \[
   a_{n+3} = a_{n+2} + a_{n+1} + a_n
   \]

2. We are given the values:
   \[
   a_{28} = 6090307, \quad a_{29} = 11201821, \quad a_{30} = 20603361
   \]

3. We need to find the remainder when \( \sum_{k=1}^{28} a_k \) is divided by 1000. Let \( S = \sum_{k=1}^{28} a_k \).

4. Notice that the sequence is defined recursively, and we can express \( S \) in terms of the given values. We can use the recursive relation to find a pattern or a direct formula.

5. We observe that:
   \[
   a_{n+3} = a_{n+2} + a_{n+1} + a_n
   \]
   This implies:
   \[
   a_{28} = a_{27} + a_{26} + a_{25}
   \]
   \[
   a_{29} = a_{28} + a_{27} + a_{26}
   \]
   \[
   a_{30} = a_{29} + a_{28} + a_{27}
   \]

6. We can sum the first 28 terms:
   \[
   S = a_1 + a_2 + a_3 + \cdots + a_{28}
   \]

7. Using the recursive relation, we can express \( S \) in terms of \( a_{28}, a_{29}, \) and \( a_{30} \):
   \[
   S = a_1 + a_2 + a_3 + \cdots + a_{28}
   \]
   \[
   S = 1 + 1 + 1 + (a_4 + a_5 + \cdots + a_{28})
   \]

8. We know that:
   \[
   a_4 = a_3 + a_2 + a_1 = 1 + 1 + 1 = 3
   \]
   \[
   a_5 = a_4 + a_3 + a_2 = 3 + 1 + 1 = 5
   \]
   \[
   a_6 = a_5 + a_4 + a_3 = 5 + 3 + 1 = 9
   \]
   and so on.

9. We can use the given values to find the sum:
   \[
   S = \sum_{k=1}^{28} a_k
   \]

10. We can use the fact that:
    \[
    a_{n+3} = a_{n+2} + a_{n+1} + a_n
    \]
    to find a pattern or a direct formula for the sum.

11. Given the values \( a_{28}, a_{29}, \) and \( a_{30} \), we can find the sum modulo 1000:
    \[
    S \equiv \sum_{k=1}^{28} a_k \pmod{1000}
    \]

12. We can use the given values to find the sum modulo 1000:
    \[
    S \equiv 6090307 + 11201821 + 20603361 \pmod{1000}
    \]
    \[
    S \equiv 307 + 821 + 361 \pmod{1000}
    \]
    \[
    S \equiv 1489 \pmod{1000}
    \]
    \[
    S \equiv 489 \pmod{1000}
    \]

The final answer is \( \boxed{489} \).