Analysis: Since $a^2=97344$ and $c^2=98596$, by using the difference of squares formula, we get $(c+a)(c-a)=1252$. Given that $a$, $b$, and $c$ are three consecutive positive integers, we have $c-a=2$ (1). Therefore, we can calculate $c+a=626$ (2). From (1) and (2), we can solve for $c$, and thus find the value of $b$.

Let's solve it step by step:

1. From the given, we know $a^2=97344$ and $c^2=98596$. Using the difference of squares, we have:
$$(c+a)(c-a)=c^2-a^2=98596-97344=1252$$

2. Since $a$, $b$, and $c$ are consecutive integers, we have:
$$c-a=2 \quad \text{(1)}$$

3. From the equation above, we can find:
$$c+a=626 \quad \text{(2)}$$

4. Solving equations (1) and (2) simultaneously, we can find the values of $c$ and $a$, and thus determine $b$ as the integer between $a$ and $c$.

Therefore, the value of $b$ is $\boxed{313}$.