It sounds to me like one's best bet to deal with a revenant (assuming you're a target and don't wish to be) would be to contain it. Place the revenant in some situation that it cannot escape-- at least, not until the person who they want to kill is already long gone from natural causes.
To that end, I'd imagine that you have people who would establish businesses where they would attempt to prevent the revenant from accomplishing its goal. As Starfish Prime stated in their answer, it might be a good idea to cast the revenant in concrete after blowing it apart, depending on how they work when blown up. They might lock a revenant in an iron vault of some sort (since iron is said to have powerful properties against the supernatural in some folk lore). The idea here is that the revenant might be able to wear through the metal through time and energy expended-- but it will almost certainly take decades to do so-- at which point, the revenant's target will have already passed.
I think that it would make crime a little bit less personal, with a focus on anonymity. If I'm going to do something sinister (and there will always be people who do something sinister), I would want to disguise myself. If I can prevent you from knowing my identity, that's kinda useful (the revenant would only know who to look for if I re-used my disguise). But if I could convince you that I was someone else entirely while I did something sinister? Maybe even an enemy of mine? Why, I could kill two birds with one stone, potentially!
Example scenario: Jane Q Evil is a vindictive and greedy lass. Her half-sister, Becky J Good stands to inherit the complete family fortune when their parents die, and Becky already has a family and children. Jane knows that her sister would leave all of the money to her husband and children, but never to Jane. Jane also passionately hates her stepmother, believing that her father made a mistake with that marriage.
Jane disguises herself as Becky, then starts to do innocuously crappy things to her stepmother. Nothing too huge-- stepping on her dress without apologizing, stealing a few dollars from her, etc. When the stepmother tries to confront the real Becky, Becky obviously denies it and says that that didn't happen. The stepmother is now starting to really dislike Becky, as it seems that she's gaslighting her. This is the most dangerous part of the plan-- if Becky ever figures out that it's Jane disguised as her, Jane's got a potential revenant on her hands. But if Jane can convince Becky that their dear old step-mother is starting to go senile, she's got a great situation on her hands.
Jane, disguised as Becky, murders her step-mother one night. She makes sure that the step mother gets to see that it was "Becky" who killed her, and that she knows that the murder was purely out of greed and the desire to swindle for money. In short, she ensures that her step-mother becomes a revenant.
A few days later, the step-mother comes back and murders Becky. Jane's hands are clean in that murder, where she would most clearly stand to benefit. Now Jane can just wait for her father to pass and rake in the dough.
Of course, if Becky's suspicious of something, she might order Dear Old Mom's corpse to be placed in a cast iron safe filled with concrete and buried in quicksand that her mother couldn't crawl out of as a revenant. That would put quite the hitch in Jane's plan!
Edit: Your post asks for culture, and while crime is an aspect of culture, I'm definitely not answering your question completely.
I think that different groups might view revenants differently. Ordinarily, assuming all is well, it might be super common to put all corpses into a standard burial practice where a revenant couldn't easily escape (IE the iron coffin caste in concrete). But what about a case where there's an unsolved murder? The town might take a vote to leave the corpse free so that it might solve the murder and kill a murderer in their midst.
One other thing I think you'd find-- there wouldn't be such a thing as slavery in this world. 1800's America thrived on the cheap labor of indentured servants (who were hit with wildly unfair labor contracts) and the free (minus food and housing costs) labor of slaves. Both groups would be extremely inclined to come back and kill the people who used them for their entire lives. Workers probably have really, really good compensation in this world; a mogul can't afford to demand 15 hours days 7 days a week from people who might come back and murder him in a few year's time.
There's probably a strong inclination to respect the elderly, and a ton of disrespect from the elderly toward those younger than them. Consider-- if grampa dies hating me, he probably comes back and kills me while I'm still relatively young (in my 20s or 30s). But if I'm grampa and I'm a completely terrible human being to my grandchild-- I'm dead by the time that kid passes! What do I care if the kid was upset enough with me to become a revenant?
Doctors may be extremely scarce in this world (and those who are around are highly revered). Medical practices might also not have advanced beyond what seems obvious right now. If a doctor does something that people view as controversial (say, creates an early vaccine with scrapings from a scab that comes from an infectious disease) and a patient dies (whether because of what the doctor did or not), there's a decent chance that the patient or their relatives/dependents will interpret this as being the doctor's fault. If someone believes that the doctor is responsible for their passing and they left a dependent family behind, they might come back as a revenant and kill the doctor.