One-step majority-logic decoding is one of the simplest algorithms for decoding cyclic block codes. However, it is an effective decoding scheme for very few codes. This paper presents a generalization based on the "common-symbol decoding problem." Suppose one is given M (possibly different) codes over the same field; suppose further that the codewords share a single symbol in common. The common-symbol decoding problem is that of estimating the symbol in the common position. (This is equivalent to one-step majority logic decoding when each of the "constituent" codes is a simple parity check.) This paper formulates conditions under which this decoding is possible and presents a simple algorithm that accomplishes the same. When applied to decoding cyclic block codes, this technique yields a decoder structure ideal for parallel implementation. Furthermore, this approach frequently results in a decoder capable of correcting more errors than one-step majority-logic decoding. To demonstrate the simplicity of the resulting decoders, an example is presented.