Going after the murderer - in this case, a serial killer - is a logical move on this chess board. But in time travel, not only is time itself your ultimate opponent, but so is a perpetual lack of crucial information. It's hard enough to know all the relevant factors - in the case of police work, identifying the correct suspects - but when you're doing this across time it gets even harder.
So if you're familiar with the playbooks of time travel, it's no big surprise that Raimy and Frank, after zooming in on and chasing the guy they were increasingly sure was the serial killer, across two episodes, discover that he's not. But the payoff was nicely done, with the guy disappearing literally before Raimy's eyes, as he's hit by car as he's trying to run away from Frank in the past.
In a way, this is a good metaphor for Frequency and all good time travel: you can't run away from your fate. This could be the theme of any powerful story, but in Frequency, the tension is heightened by our not knowing exactly what that fate is. As I've said in reviews of earlier episodes, Frequency is a story not about major, world-changing events but about a family, a daughter and her father, and this makes it all the more appealing.
One more thing: here's my prediction for who the killer is: I think it's Raimy's neighbor. Not that he's done anything wrong, but there's something about the way he keeps popping up that's making me think he's up to no good.
See also Frequency 1.1: Closely Spun Gem ... Frequency 1.2: All About the Changes