Your question is a little unclear. The circumstances in which the State can appeal an adverse decision in a criminal case are very limited. An appeal by the State from a jury's verdict or a judge's finding of not guilty is, of course, not permitted. In many states, however, the prosecution can appeal a preliminary legal ruling that impairs the State's abillity to present its case. An obvious example is a ruling granting a defense motion to suppress evidence due to an allegedly unlawful search and seizure. If the judge's ruling on the motion to suppress makes it impossible for the State to proceed effectively with the prosecution, it may be able to appeal the ruling. This is a very technical area and every case would have to be addressed individually and in light of applicable state law.
In addition to the legal question of whether the State can appeal a decision there is the practical question of whether it wants to bother. If the case is large enough or the issue important enough, it may. If the case is less important, if the judge's decision has a good chance of being upheld, if an adverse ruling on appeal would make things worse for the State in other cases, or if the prosecutor just has more urgent matters to attend to, the State may just let it go.
Ohio actually affords broad appellate rights to prosecuting attorneys. Prosecutors can indeed appeal not guilty verdicts, although retrial is obviously barred by the Fifth Amendment of the Constitution as well under Article I, Section 10 of the Ohio Constitution.
Your question doesn't indicate who "they" are who looks "through so many lower court decisions." Are you trying to determine what criteria the Ohio Supreme Court uses before accepting a discretionary appeal?