In this question it was shown that it is not possible to bypass the States Attorney before the grand jury. But what relief exists in cases where Prosecutorial Discretion might be used to protect people who are powerful, influential, and/or friends of the Court?
I have encountered this problem in Pennsylvania. The PA Code requires a District Attorney to approve all private criminal complaints. If the DA declines to prosecute, then an affiant can petition the Court of Common Pleas to review the decision. However the affiant bears the burden of convincing the court that the DA abused his descretion in declining to prosecute, which is a pretty high hurdle.
In the United States the only other legal appeal I am aware of is through federal courts under broad federal laws like 18 USC 242 or 42 USC 1983.
As @feetwet points out, there are a few jurisdictions (New Hampshire and Pennsylvania are the only two I'm aware of) where a private citizen may initiate a criminal prosecution. However, these are limited and rarely used procedures, and are not available in most jurisdictions.
If someone has committed a crime, and in so doing has caused you a legally cognizable injury, you may also file a civil lawsuit. Many criminal acts also create tort liability in favor of the victims of the crime; for example, wrongful death suits are often filed in homicide cases.
In Virginia, it is possible to initiate a criminal prosecution for a misdemeanor by obtaining an arrest warrant from a Magistrate.
Code of Virginia § 19.2-71 (emphasis mine)
§ 19.2-71. Who may issue process of arrest. A. Process for the arrest of a person charged with a criminal offense may be issued by the judge, or clerk of any circuit court, any general district court, any juvenile and domestic relations district court, or any magistrate as provided for in Chapter 3 (§ 19.2-26 et seq.) of this title. However, no magistrate may issue an arrest warrant for a felony offense upon the basis of a complaint by a person other than a law-enforcement officer or an animal control officer without prior authorization by the attorney for the Commonwealth or by a law-enforcement agency having jurisdiction over the alleged offense.
As you can see by the above statute, a felony warrant can only be issued on the request of a "law-enforcement officer or an animal control officer" unless prior authorization "by the attorney for the Commonwealth or by a law-enforcement agency having jurisdiction over the alleged offense" has been obtained.
Code of Virginia § 19.2-37 sets out the requirements for being appointed as a Magistrate, and none of these include being an attorney or licensed to practice law.
This procedure for obtaining action on a Misdemeanor is commonly known as "swearing out" a warrant. The Magistrate is not required to issue the warrant, but they may do so, without needing to call in a law enforcement officer or District Attorney.