I believe that the Texas/Southern U.S. version of the expression is "I don't have a dog in this [or that] hunt," meaning that the speaker doesn't have a rooting interest. The expression is sometimes used to emphasize the disinterested status of the speaker prior to his or her offering advice or commentary on the subject of the "hunt" in question. The speaker's claimed lack of any ulterior motives as an observer supposedly enhances the reliability of the advice or comments offered.
The expression "I don't have a dog in this hunt" sounds folksy enough to be a genuine old-timey expression, but a Google Books search suggests that it is a recent, faux folksy saying, along the same lines as calling someone "all [cowboy] hat and no cattle." Google Books finds one (relatively) early example from 1994 and another from either 1994 or 1995 (judging from the references it makes to O.J. Simpson's impending murder trial, which took place in 1995). From U.S. Congress, Senate Committee on Small Business, Subcommittee on Regulation, Business Opportunities, and Technology, Problems in Securing Informed Consent of Subjects in Experimental Trial of Unapproved Drugs and Devices (May 23, 1994):
Mr. COMBEST. I had one more question. We have a saying in Texas that I don't have a dog in that hunt, which I don't. I don't have a bias in this one way or the other. We are just trying to find out the total details so we can come up with hopefully some positive suggestions.
From Instauration, volumes 19–20 (1994–1995) [combined snippets]:
How many Majority folk are players in this game? Well, there's Nicole. Mmmm, who else is there? The only other Majority female I'm aware of is Judge Ito's white wife. Everybody else is either black or Jewish. If you're a white male, you don't have a dog in this hunt. Might as well just sit back and enjoy the spectacle.
The next Google Books match is from U.S. Congress, House Committee on Government Reform and Oversight, Subcommittee on Civil Service, Government Shutdown One (1997) [combined snippets]:
I thank the distinguished chairman and members of the subcommittee. I will submit my written testimony and try to paraphrase it. Like Mr. Browder, I don't have many Federal employees. I'm not near the nation's capital. So, in one sense, I don't have a dog in this hunt. But in the sense that I represent 600,000 taxpayers, I think I do have a reasonable reason to be somewhat concerned about it.
These hearings took place on December 6 and 14, 1995, and the record of the hearings was published on January 1, 1997.
None of my American slang reference books from the 1990s has an entry for "I don't have a dog in this [or that] hunt." But Google Books lists many instances of the phrase from the past fifteen years.