Michael Curry had appealed his conviction, saying the case was built on circumstantial evidence that failed to prove his guilt.
The court said in a unanimous decision released Monday that the evidence was enough for the jury to find Curry guilty beyond a reasonable doubt.
The decision upholds Curry's three consecutive life prison sentences.
Curry was convicted in 2011 after cold-case investigators took a fresh look at the killings.
Prosecutors say Curry hacked his pregnant wife Ann to death, and also used the bush ax to slay his 4-year-old daughter Erika and 1-year-old son Ryan in Columbus.
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