From Steele v. Steele (Ky. Ct. App. Mar. 5, 2010), quoting the trial court:
Upon being asked by the Court to swear that he would testify truthfully, Mr. Steele declined, stating that his religious beliefs prevented him from swearing. He was then asked to affirm that his testimony would be truthful and he again declined for the same reasons. He was then asked by the Court how he wished to proceed and, in a very uncharacteristic moment, declined to express any opinion on the issue. The Court then asked Mr. Steele to simply state that he would answer counsel’s questions truthfully. Even this Mr. Steele declined to do.
Unsurprisingly, this apparently led to the judge’s not letting Steele testify; and I expect that it rather soured the judge on Steele more broadly. People who affirm rather than swearing are generally accommodated without legal objection, and I suspect without much prejudice against them as a result (though that’s hard to tell). The Constitution itself expressly provides for this particular accommodation, which Quakers (among others) have long sought. But if you say your religion tells you that you can’t swear, or affirm, or even just say that you’ll answer truthfully, Caesar is going to take a dim view of you.